Is the price difference between 1600 and 2133 even that big? I do more than gaming myself as well so that's why I try to go for the upper end.
You have to look at both the speed, and the latency
1600 mhz 7-7-7-18 might be the same price as 2400mhz 11-13-13-31, but if you want 2400mhz with the lowest latency possible, it's going to cost you. A lot.
On Newegg, for example, the lowest latency 2400mhz 16GB of ram is 9-11-11-31 and it's $269, whereas you can get 11-13-13-31 for around $100. 16gb of 1600mhz DDR3 tends to run in th $75-$150 range depending on the quality/latency.
So yeah, on the poor end of the latency spectrum, not much difference in price ($75 vs maybe $90), but if you want the best latency possible, you're going to see huge price differences. Lower latency gets harder and harder to do, the faster you go. Hell, old ass original DDR memory at like 400mhz was something like 2.5-3-3-5 latency on a 512mb chip, lol.
All that being said, the odds of overlocking your RAM are better, the lower latency your base speed is. There's a decent chance that you could overlock 1600mhz RAM that comes at a really low stock latency to a better latency 1866mhz than just buying mid-range 1866 outright.
It's definitely a balancing act. Honestly, I just shoot for the middle of the latency range when I'm buying RAM, paying a premium for super low latency doesn't do much for you, but I don't want some wonky high-latency RAM that is barely stable at the speed in which it's running.