The closest you can get to a general purpose drill bit is a basic twist drill made for metal. That works equally well on softer materials like wood and plastic. Other than that, no. You typically choose the bit based on what you're drilling into. Also, not all twist drills are made equal. HSS-R (rolled) works for wood, plastic and soft metals like aluminium, copper or brass. HSS-G (ground) is the bit you probably should be using especially if you're drilling into steel. Then there's also HSS-Co or HSS-E (cobalt) which is for hard metals like stainless steel. Then there's also carbide bits but those are for drill presses and such.

Diamond bits work the way you describe but they're for ceramics and glass. If you try using one on wood or plastic it'll just clog up immediately. There's a reason twist drills look like that: they're designed to lift the removed material from the hole so that the bit can drill. A bit similar to how an orbital sander works much better with a vacuum attached so the dust layer doesn't sit between the sanding disc and the surface you're sanding.

Then again, for concrete you not only need a masonry bit but ideally one with an SDS+ attachment for a dedicated hammer drill. You could technically use a diamond or non-hammer masonry bit as well but it would be extremely slow and you'd probably overheat the bit which affects the heat treatment of it which then causes it to get dull and quit drilling and just start heating up even more.