How to fix a toilet that flushes incompletely

If the problem is mineral clogging of the under-rim channels on the toilet, and this does produce the symptoms you describe, then neither CLR no vinegar may be enough to clear up the blockage. The traditional treatment is a cup or two of muriatic acid (HCl 36%, from the hardware store) down the overflow hole in the tank. That goes directly to the rim vents. Pour the stuff in, wait an hour, and flush. Gloves and eye protection are recommmended, and a funnel so as to avoid spillage and corroding the metal parts in your tank. This looks like decent writeup of the method: How to Fix a Slow Toilet

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.