Custom, from scratch processor design - is it possible?

"Ask Intel to create a chip ..."

You can ask all you want. You will be ignored unless you show up with a check in hand for, say, US$100M engineering cost and a commitment to buy US$1000M parts over say 10 years time.

Why exclude FPGAs? You can do your own full custom processor design today and fit it in a 500+/-pin FPGA and cost way less than the above option. You can easily do it on a US$500 high end FPGA development board.

The NRE cost today of doing a full custom design is well beyond the limit of any hobbyists, except maybe Elon Musk or another hobbyist billionaire.