I tried using the .apkm (installed with APKMirror app), instead of the .apk, of version 20241120_00_RC07, which has worked the best for me, in terms of quickly connecting to RCS and working every time. I was hoping that this might solve the problem of the old version of Messages not connecting to Etouffee. But no dice. I still had to update to the Google Play version to get Etouffee.
Next time, I'll try the .apkm for 20240708_03_RC04 that @Pixelated said does connect for them to Etouffee, without having to update to the Play Store version. And maybe I'll try 20241120_00_RC01 as well, that at least one person said they had luck with.
One advantage of using the .apkm, I discovered, was that when I went to set the old version of Messages as the default SMS app, I did not have to enable restricted permissions. Instead it was immediately available to be set as the default. On the other hand, updating to the Play Store version took a weirdly very long time (I thought it was stuck and failing in some way). Play Store updates have always been slow, but this was even slower.