BTW I do like the idea of a “dumb” key that can just unlock the door if you lock a primary key in the car.
For modern cars you can get transponder key blanks off eBay for less than $20 (and sometimes less than $10). These are essentially valet keys with an uncut blank and an un-programmed transponder, but no keyless entry device. Here’s an example.
Or, there are blanks with keyless entry shells without the guts, like this one.
I think it would cost another few bucks to have a locksmith cut the blank to work in your door locks. It would cost additional to have someone program the transponder to work with your car’s security system, or you could do it yourself (tons of guides online). Or you could just hack off the transponder part and have the key be more in line with what the OP was proposing. I’d still want a little nub of plastic to give me leverage on opening doors.
One thing I noticed is that, even if the item is advertised to work with a specific make/model/year, the key blank may not be compatible. For instance, my key blank is stamped “V” but I can only find blanks with stamps “S” or “F” on eBay. So caveat emptor.

