> Unfortunately older machines just don't have the processing power to handle HTTPS

Is anyone aware of the nature and extent of the limitation here? Are we talking minutes or years? Would more RAM help?

No, it's largely CPU bound. I had a 68040 that would take around 45 seconds to generate ssh keys, and my little NetBSD Macintosh IIci with a 25MHz 68030 and no L2 card took 22 seconds for a single short TLS 1.2 transaction to a local test server. This wouldn't be a big problem if it weren't for the fact many servers just don't wait.

Shameless plug: a crypto client library oriented to old things. https://github.com/classilla/cryanc