Toilet tank not filling

That's a fairly popular fill valve. To open it:

1. Shutoff the water, you don't want a geyser.
2. Lift the float to the top position.
3. While holding up the float, push down on the top cap and twist the cap a quarter turn (I believe counter clockwise).

The cap will then lift off and you can inspect and clean any parts. Before reinstalling the cap, move any small parts to a safe location, get a small cup to place over the top of the valve, and turn on the water supply. This will jet water straight up from the top of the valve and clear out any debris.

Reinstalling the cap is the reverse of the above procedure. Put the parts back, position the cap back on top, lift the float, press down on the cap, and twist.

Note that if cleaning it doesn't work, it's fairly simple to replace this fill valve.

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.