BFG-9000
Splendid
Well it is fortunate that you can get exactly that nowadays for $18 in something like the Thermalright Assassin X120, isn't it? That's cheaper than OP's present cooler so definitely qualifies as a cheap cooler, and is good for 125w TDP, which is good enough for a 8700 non-K, especially when not using its IGP.No. Do not get a cheap crap cooler.
i7-8700 requires a decent heatpipe cooler. Minimum 4 heatpipes and 12cm fan.
Stepping up to their $33 twin-tower Peerless Assassin gains 245-265w TDP capability which should be quieter and a much better deal than the similarly-priced usual budget suggestion of a single-tower Hyper 212. I'd expect OP would be quite leery of buying from a crap company like Cooler Master or Thermaltake too given their recent experience, and Thermalright have long had a solid reputation.