I just found this old-ish unanswered question, so thought I should take a stab at it.
In short - is this approach valid/safe?
It all depends on what you plan to do with them.
If you are just using it in server-side code, then yes. You simply store the Id of the timezone, and show the user the corresponding DisplayName. FindSystemTimeZoneById and GetSystemTimeZones are perfectly valid for this.
Keep in mind that while the Id values are always the same - the DisplayName properties will be different based on the language of the Windows operating system that you are running the code on. The implementation is not culture aware, so just setting a different target culture in .Net will not change the DisplayName string.
The records in the Microsoft Windows time zone database are reasonably stable, and are kept updated via Windows Update. Some of the information comes from the registry, but the localized resource strings come from tzres.dll. Of course, all of that is hidden from you by TimeZoneInfo and related classes.
However, if you are passing these time zone Id values to other systems - watch out. There are some variations with prior versions of Windows. For example, I know that the display names used to all say "GMT" in them, and now more correctly say "UTC". The Id values are the same, but who knows exactly what else is not consistent. Especially if the target computer hasn't received the same set of Windows Updates that you have. By the way - the updates are announced here.
You should also know a few things about the Windows time zones database:
It is incapable of representing more than two DST transitions per calendar year. For example - in 2010, Cairo, Egypt had four transitions. Microsoft issued a hotfix, but that only made a compromise correction, not a historically accurate one. There are other zones that have historical changes like this that cannot be represented properly.
Some things changed in between Windows XP/2003 and Vista/2008. There is some really good information about this here, along with a lot of detail about how the registry is used.
Microsoft is the only major player with their own time zone database. The rest of the world uses the IANA/Olson database. This leads to interoperability problems. I have a decent write-up of this in the timezone tag wiki.
You should also know that TimeZoneInfo is inextricably linked to the DateTime and DateTimeOffset classes. These have their own set of quirks. DateTimeOffset is somewhat usable, but DateTime is riddled with nuance. Read:
A far better solution to date and time in .Net is to use NodaTime. This library implements both time zone databases, including the CLDR mappings between them. It also provides a much safer API that doesn't let you get into trouble. It may take a bit of re-learning, but before you know it, you will be using classes like LocalDateTime, ZonedDateTime, OffsetDateTime and Instant.
You still have the problem that the time zone database is updated frequently, as we leave timekeeping rules up to the politicians. But the TZDB folks do a pretty good job of keeping the database historically accurate, and providing aliases/links when zone names change. You can see a list of the tz names here.
Also, if you go with NodaTime, you can choose to either stick with the copy of the TZDB compiled into the distribution, or you can ship your own copy with your application. You could (theoretically) write your own code to pull down the latest version from IANA and keep your application updated - all without relying on the host operating system.