Well, I'm almost at 72 with no reset and RCS continuing to work.
I was looking again at what anorak said worked for them. One of their steps was to allow READ_DEVICE_IDENTIFIERS for Google Play Services. I had not bothered to try this, even though it has been discussed a lot over the course of this thread, because my understanding was Google Play Services should be getting this information anyway (although I'm not totally sure about this). But then I was looking again at the bug report for this issue in Graphene, https://github.com/GrapheneOS/os-issue-tracker/issues/6331, which explicitly states the problem has to do with Messages not being able to READ_DEVICE_IDENTIFIERS. So I thought I may as well try this.
What I did was:
1) After the 36 hour reset with Messages, but before installing 20241120_00_RC07, I set READ_DEVICE_IDENTIFIERS to allow for Google Play Services.
adb shell appops set com.google.android.gms READ_DEVICE_IDENTIFIERS allow
I don't honestly remember if I had already uninstalled Messages or just did this before I started any of the steps of the 36 hour reset process. Sorry.
I did not set READ_DEVICE_IDENTIFIERS to allow for com.google.android.ims (Carrier Services) as @anorak mentions, because my understanding is that Carrier Services is not the issue and I don't have Carrier Services installed anyway. @ZenIsFluffy's experience a few posts up, regarding Carrier Services, is interesting, but difficult to tell which thing they did actually helped. Also I wanted to only change one thing at a time, to be scientific about this.
2) After that I installed 20241120_00_RC07 as usual, set is as the default SMS app, and got to "connected."
3) Then I updated to 20250311_04_RC01 as usual. I did notice that it immediately got Etouffee to "true," rather than having to reboot a couple times, as I have had to lately.
That's it. I did all this almost 72 hours ago.
I do wonder if this means that if I updated to the Play Store version it would keep working. But I'm also hesitant to jinx anything. Separately, a few people said that the 36 hour reboot started again after a recent update to Graphene. I do wonder if something changed with permissions on READ_DEVICE_IDENTIFIERS and that brought the 36 hour reset problem back.
Anyway, I hope this works for other people and helps.