What is the video equipment the church has?
If we are talking about simple 1080p HD video you can buy any modern intel CPU that has an iGPU, even a pentium, and the iGPU will be more than enough to capture that and turn it into an x264 stream, actually it doesn't even turn it into that it just captures it like that in the first place.
It's called quicksync and has been a part of intel iGPUs for the whole lineup since haswell.
You can also use nvidia and AMD GPUs for that (or even one of intel's new ARC GPUs) but an extra GPU is extra cost for no reason.
CPU encoding is something that some crazy people do for max quality while playing games with very fast movements where the iGPU method might introduce a little bit of pixelization if you don't use a high enough bitrate, you won't need that for church events.