Yes, if your server is a weak, limited processor, you want to keep the demands on it as low and lean as possible, and let the client do the heavy lifting. HTMX is not a good fit for this scenario, just like PostgreSQL is not a good database to embed on your devices.
This isn't a controversial idea and nobody would try to sell you on HTMX for your use case.
1. No, templating strings is actually quite cheap. I'm doubtful that you could benchmark any substantial difference between templating html and serializing json.
2. Who has a server with a weak, limited processor? HTML templates power Django, Rails, and PHP. This paradigm worked fine on the servers of 20 years ago, in the slowest languages we use. I could serve a Django app on my phone and see reasonable performance.