Slow filling toilet tank after new valve kit installation

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.

Usually you can adjust the float so that it shuts off at a lower water level. Also, sometimes the float is just getting stuck (and it needs to be cleaned).

Edit: If adjustment or cleaning does not work, then you need to replace the fill valve (it's not an expensive part).

Edit2: Reiterating Ed Beal's comment- you may want to start with turning off the water to the toilet and flushing the water out. Then you can inspect or replace the fill valve. A fill valve kit can currently be had for under $10.