If I snapped the head off of a 1/4" bolt with the correct TPI trying to get it in the hole, I'll do the same with a 1/4" self-tapper. The issue is that a 1/4" shank is too big.
I like the idea, but I'm not seeing 7/32" self-tappers. Be willing to try, though, if you have a link.
What is a weird limited-application screw in 2024 might not have been 61 years ago. Just sayin'.
In rough finished holes, they won't go to 1/32" increments. The self-tapper sizes the hole correctly for the thread, where your existing hole won't fit a 1/4. I'm operating under the assumption that these are threaded in sheet metal.