Update: I fixed it. See answer below - Visual Studio 2015: Multiple package load error after Update 3 failure: Windows 10
So that's that.
- Visual Studio 2015 with Update 2 was installed on Windows 8 64 bit. Everything was working well and good.
- Windows 8 was upgraded to Windows 10. Visual Studio still worked, no issues.
- While adding ClickOnce tool via setup, Update 3 got installed as well.
 Everything went and hit the proverbial fan.
Activity log XML is as below:
461 ERROR Exception parsing Advertised (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.143 
462 ERROR Exception parsing AssemblyFolders (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.143 
463 ERROR Exception parsing Fusion (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.143 
464 ERROR Exception parsing NGen (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.143 
465 ERROR Exception parsing NGenQueue (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.143 
466 ERROR Exception parsing policy (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.143 
467 ERROR Exception parsing Windows Presentation Foundation (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.144 
468 ERROR Exception parsing Advertised (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.144 
469 ERROR Exception parsing AssemblyFolders (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.144 
470 ERROR Exception parsing Fusion (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.144 
471 ERROR Exception parsing NGen (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.144 
472 ERROR Exception parsing NGenQueue (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.144 
473 ERROR Exception parsing policy (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.144 
474 ERROR Exception parsing Windows Presentation Foundation (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.152 
475 ERROR Exception parsing Advertised (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.152 
476 ERROR Exception parsing AssemblyFolders (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.152 
477 ERROR Exception parsing Fusion (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.152 
478 ERROR Exception parsing NGen (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.152 
479 ERROR Exception parsing NGenQueue (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.152 
480 ERROR Exception parsing policy (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.152 
481 ERROR Exception parsing Windows Presentation Foundation (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.154 
482 ERROR Exception parsing Advertised (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.154 
483 ERROR Exception parsing AssemblyFolders (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
484 ERROR Exception parsing Fusion (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
485 ERROR Exception parsing NGen (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
486 ERROR Exception parsing NGenQueue (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
487 ERROR Exception parsing policy (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
488 ERROR Exception parsing Windows Presentation Foundation (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
489 ERROR Exception parsing Advertised (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
490 ERROR Exception parsing AssemblyFolders (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.155 
491 ERROR Exception parsing Fusion (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
492 ERROR Exception parsing NGen (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
493 ERROR Exception parsing NGenQueue (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
494 ERROR Exception parsing policy (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
495 ERROR Exception parsing Windows Presentation Foundation (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
496 ERROR Exception parsing Advertised (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
497 ERROR Exception parsing AssemblyFolders (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
498 ERROR Exception parsing Fusion (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
499 ERROR Exception parsing NGen (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
500 ERROR Exception parsing NGenQueue (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
501 ERROR Exception parsing policy (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.156 
502 ERROR Exception parsing Windows Presentation Foundation (under HKEY_LOCAL_MACHINE\Software\Microsoft\.NETFramework) as a version   Microsoft.VisualStudio.IDE.ToolboxControlsInstaller.ToolboxInstallerPackage 2016/08/02 09:38:18.157 
503  Done loading new toolbox content, total time = 265 ms 
