There are a whole bunch of different problems, reasons, solutions, wear, time, wood, etc. happening at that end of the bow, so let's look at a few.
It sounds like the eyelet (the brass/bronze part in the frog that the screw goes through to connect the frog to the stick) was loose, and you tightened it as far down as you could before it started to bind. That's the right thing to do, and can/should have been checked during the rehair. They sometimes start to wear, get a bit loose, the stick shrinks and the wood wears inside the holes. If everything is properly aligned that shouldn't be a problem, but if the hole in the back of the stick isn't centered and in line with the smaller hole at the front, a gap can open up here.
The octagon (or some other 8 sided shape with different angles) of the stick and the frog both have to be perfectly the same where they fit together and the silver underslide acts as a bearing surface on two mating planes, but often they aren't cut perfectly the same, the stick is slightly smaller, or maybe the line wiggles a bit, isn't perfectly flat, leans a bit, etc. If it's handmade, on a bass bow that's a lot you've got to get perfect to make all the gaps go away. If this isn't a perfect fit, it also can get really full of "gunk" from rosin, your hands, whatever the rehairer lubricated the screw with, metal dust from the eyelet slowly wearing out because the gap usually means things aren't quite right, etc.
Should the button and the stick be in line with each other? Yes. Are they always? No. Some makers are drilling those holes by hand and cutting the octagon by hand and don't get either of those things perfect, some do it at various stages during the process, and everything might be aligned earlier, and then not later, and sometimes, they just aren't perfect. If you take the screw out and look at the end of the stick, is the hole in the middle? There should be a nipple at the end of the stick too, which may or may not be centered as well, and a part of the solution or problem.
All of it can be corrected. If it wasn't like that before the rehair, then it should be made right, but if the holes weren't drilled in the right place to begin with, unless the rehairer started messing around with those, the bow could have any of the above issues or a combination of them that end up with it being not quite right back there, but especially on German frogs where the contact between the frog and the stick is so long, it's pretty common for a small gap to happen. Sometimes you just need to turn the eyelet, sometimes it needs bushings, sometimes it's a whole thing.