Why did my roast beef turn out chewy and not tender? Where did I go wrong?
The only mistake you made was the choice of cut, and maybe the quality of the beef itself. Round (in the UK/AU/NZ topside and silverside) is from the rear end of the animal, and is a working cut. Working cuts have to exert a force, so the muscle must have lots of collagen to distribute the force from the tendon throughout the muscle. Collagen is a tough material which breaks down in the presence of heat and moisture, not just heat, so working cuts make poor roasts or steaks and are better braised or stewed. 

The upside to working cuts is that they have a stronger flavor, and the collagen when broken down by moisture turns to gelatin which adds to the flavor and mouth feel. A well-prepared braised piece of beef is as good as a roast any day in my opinion. It's also better value, round and other working cuts are much cheaper than tender cuts. 

Your technique as described is perfect, and if you'd chosen a rib, short loin, or sirloin roast (US cuts) it would have come out beautifully. Where you erred was at the store when you chose a working cut for roasting. 

Now some people on the forum are probably preparing a rebuttal at this point saying "you can roast round and have it be tender", and they'd be right up to a point. The best quality beef is reared and slaughtered better, so if you bought US Prime round it has a good chance of coming out relatively tender, however it's hard to find. My assessment would be for what you'd find in the average store in the US, which is Choice grade. Choice encompasses something like 75% of beef produced, so has a huge range of quality. 

You can tell a lot about meat by touching it. Stick your fingers in it, if they go in easy and the meat springs back then you have a tender cut, if you can't get your fingers in it it's a braising cut. If you stick your fingers in and the meat doesn't spring back its old, so don't buy it.