Does he just...never want to?
If you have seen Solo, L3-37 has the brain of an R3 astromech, who wasn't able to speak Galactic Basic, but salvaged parts from other former droids and now can. But she also upgraded her mind, including memory and processors, so it's unclear which component gave her speech, or it is none, and she altered her own programming as well.
I try to imagine it as being similar to a human who is unable to vocalize a human language, though they understand it perfectly fine, and can communicate through other means. Except in the cases of the droids it is probably done on purpose by those programming them. And in addition, they may also lack the hardware that droids programmed to vocalize posses, as for some reason I expect that to be different than the hardware they use to play audio back.
On the surface, it doesn't make sense, but there's nothing that prevents you from building a droid that way. A decision was made to do it that way at one point, the reasoning may be now forgotten, it may have even been a mistake, or it may have been that they were designed before speaking was a thing droids could do, but it stuck. And perhaps the astromechs are smarter, so people feared a rebellion, should they be able to speak to everybody, thus all suggestions to upgrade them to have proper speaking functions were rejected.
If you see L3 and imagine at least some R3 droids having that personality and thinking, you'd see at least one reason for a droid manufacturer not to alter the way they work.
I believe he actually did speak English in the first few versions of the script for the original movie. I read the comic based on Lucas's first draft script, and I'm pretty sure the version of R2 in that spoke English.
I haven't seen the actual scripts, so I may be wrong on that one, but I would imagine R2-D2, Chopper and such, as well as everyone speaking anything other than Galactic Basic, have their lines in English originally in the script, and that's translated later on, in their cases to beeps.
Chopper is the only one who says “I've got a bad feeling about this”, and does often, in Rebels.