Plumbing – Toilet tank fill tube

I'm guessing the valve assembly inside the toilet is finding a resonance when it's just about to close. You can replace it, and it's not too difficult of a DIY job. They sell just the valves:

But if you're going to take it apart to replace the valve, I tend to buy the full kit and replace all the guts.

Note that when my toilets started to leak, it was because of high pressure in the water lines. So it wouldn't hurt to pickup a pressure gauge that screws onto a hose faucet to make sure you don't have a bigger issue. After I fixed one, another would start leaking next as the pressure got a little higher.

The overflow tube drains directly into the bowl, not the tank. The only way I can think that it would prevent the tank from filling is if it was cracked or had a hole in it and allowed tank water to drain into the bowl. If this was the case, you would notice the fill valve cycling if it was a slow leak or never shutting off if it were a large leak. The reason the tank fills quickly when you divert the bowl refill tube into the tank is that you are simply not splitting the water from the fill value between the tank and bowl any more.

It sounds like your fill valve assembly is the problem. The fill value has 2 outlets, one that directs into the tank, and the other that sends water into your bowl refill tube. My best guess is that something inside of the fill value got clogged with sediment, hard water deposits, or something else that restricted it. These are relatively cheap to replace and likely not worth the time to clean them out in my opinion.