When I try to connect stays at "Obtaining IP address" forever
This suggests the issue is DHCP unable to function for your repeater's WiFi network. Please clarify in the question, if I am wrong.
What am I missing?
From my personal experience, TP-Link's WDS Bridging feature does not actually bridge on the Layer 2 (Data link layer). Instead, it uses MAC Address Translation (MAT or sometimes, also known as ARP NAT) to connect devices from the repeater to the main router. The advantage of this approach (over 4-address mode) is that the main router doesn't have to support 4-address mode; hence it makes the repeater compatible with all routers.
However I am not sure if this is valid for your TP-Link Archer C8 as well. If your main router shows you an ARP table, you can check the entries in it to confirm.
How does it work only the first time?
I'm going to make a wild guess here and say it could be one of these reasons (though I could be completely wrong and the reason could be something else) -
The DHCP server on your main router uses the actual MAC address (from which it received the request) to allocate addresses. Now, due to MAT, all the devices connected to the repeater have a single MAC address on the main router; and hence only a single address can be allocated to it.
Try temporarily replacing your main router with something else and check if the issue persists.
The DHCP relaying on the repeater is flawed and it is not properly relaying DHCP requests.
You can try contacting TP-Link support, explain them the issue and ask for a firmware update.
What else can I try?
Separate DHCP servers for main router and the repeater
If you currently have the main router's DHCP server allocating, say 50 addresses, you could reduce it to, say 25 and enable another DHCP server on the repeater and make it allocate the rest 25 addresses. Just ensure the ranges don't intersect with each other and the default gateway provided by both is the correct router's address.
Use a DHCP server only on the repeater
Disable the DHCP server on the main router and make only the repeater handle DHCP for all devices.
Use the DHCP Relay function on the repeater
This usually complicates simple stuff and hence I ask you to try this as a last resort - Enable DHCP Relay on the repeater and enter the address of the main router as the server.
Where I can find more logs?
Usually there are logging settings in the web user interface of routers of various manufacturers, check those and enable logging for both routers. You might want to set the logging level to maximum detail (verbose) to help diagnose the issue.