No, a 5600X should be faster in all of today's games, since the new Zen 3 architecture boosted gaming performance (in CPU-limited scenarios) by more than 20% on average compared to Zen 2. And as far as core counts go, a 6-core, 12-thread processor like the 5600X is likely to be plenty to handle new games smoothly for some years to come.
As for the cooler, your existing one might be alright if its pump is still running properly, though it would not have included a socket AM4 mounting bracket if it's a pre-Ryzen model. As far as heat output goes though, the 5600X shouldn't be much harder to cool than a 6700K, despite having more cores.
Most people don't live anywhere near a MicroCenter, since they only have 25 locations across the US, and their processor deals are in-store only. If you have to drive hours to get one at that price, it's not a deal.
Also, the 9900K isn't exactly "better". AMD's SMT implementation is superior to Intel's, so the 6-core, 12-thread 5600X manages to perform roughly on-par with, and in many cases faster than an 8-core, 16-thread 9900K in most heavily-multithreaded applications, despite having fewer cores, and the same applies to many lightly-threaded applications as well, due the processor's higher IPC making up for its lower clock rates. What's more, the 9900K is very power hungry and puts out a lot of heat on Intel's 5+ year-old 14nm process, whereas the 5600X is built on TSMC's newer 7nm processor, making it far more efficient. As a result, it doesn't draw as much power or require nearly as much cooling to maintain reasonable temperatures under load. Personally, I would rather have a 5600X over a 9900K at the same price, largely down to the better efficiency, so it makes sense why the price of the 9900K has been getting slashed recently. While 5600X may be priced relatively high compared to recent 6-core, 12-thread processors, it typically performs more like Intel's 8-core, 16-thread models, which generally cost more. A 9900K at around $300 could potentially be a decent alternative though, since the 5600X is hard to come by at the moment.