I've had similar issues with MMS delivery issues. What I have been observing in a group chat, I think with one user who has an IPhone, is that some MMS's give me the "New MMS Message Tap to Download". For certain MMS group messages I'm consistently getting stuck. It's been happening pretty consistently with MMS's from certain phones for the past couple of weeks. SMS always seems to go through fine.
My setup is a Pixel 7A, default message app, Verizon Sim. I have tried restarting multiple times and various settings. Been using this setup as my daily driver since around October last year and this has been my one significant snag.
I did try psf's suggestion for Google Play services (which I had previously avoided installing) and it didn't seem to resolve it for me. Wondering if there were other ideas.