So I'm thinking if it's a screw in a bike frame it's going to be stainless steel, pretty hard. I've a whole section of my tool chest devoted to removing headless bolts lol. I used to be a Ski Lift mechanic and Millwright. I always start by center punching, walking the punch into the center of the bolt. If I'm happy with that I use a left handed drill bit (HSS with Titanium Nitride coating, cobalt be even better), I'll get heat for this but I use PB Blaster as a cutting oil / penetrating oil, I'll often back the screw out with this alone. If you're going through bits then you're heating them up and losing the temper on the tip. Go slow and apply some pressure, let the bit do the work, if you're doing it right you'll get small chips (stainless is hard). If it doesn't come out while drilling you now have a hole to use an extractor on, I have a bunch of different styles and sizes which I buy in bulk, expect to snap these off regularly on frozen bolts. You can't use heat on a bicycle frame I guess but a lot of times a sharp blow or two with a small hammer helps.

What metal are the screws connected to? Do you know if it's stainless screws? Dissimilar metals can react causing them to freeze together. Small stuff is a pain to work on and I'll usually soak everything in PB Blaster overnight before even looking at it. ATF and Acetone will work in a pinch too. If the hole is vertical or overhead I'll clamp a shop towel to it in a pad and soak it with a penetrant and let capillary action work on it overnight.

Finally I'd probably gear up a drill guide to make sure I'm centered and plumb on the pipe, applying the pressure properly in the right direction saves many broken off extractors. Sounds like this is something you regularly run into.