Now all you need to do is
do your SHA256 hash by hand to generate the checksum and you've got the complete no computer solution to generating a seed mnemonic

It would however blow out your 15-20 minute time frame to probably closer to a day... having to do 64 rounds of SHA256 to get your final hash at a rough speed of 1 round per 15 minutes or so

There are just some things that are better left to computers

well you don't have to have a checksum since it is not mandatory. you can just pad the entropy with zeros and then derive the mnemonic from that.
besides the problem is never with checksum and things like that. the problem that makes people want to flip coins is the Random Number Generators, everything else can still be done with a computer after the number was physically generated using a coin or something like that.