What the shop guy was talking about was getting power to it, not getting heat out. The 5700X3D doesn't make too much heat, but it does run a bit hot. Some cheaper A320/A520 boards have very low end Voltage Regulator Module (VRMs) that cannot run higher end CPUs at full power. Generally this means they struggle with the dual CCD Ryzen 9s, but all but the worst of them can at least run the single CCD chips at full tilt. Some of the really cheap ones don't even have VRM heatsinks (yours does).

The 5700X3D is single CCD and does not run high clock rates or voltages and the A520 Aorus Elite was reviewed as having decent enough VRMs, so the 5700X3D should be fine with it.

As far as cooling goes, you can slap on a nice air cooler like a Thermalright Peerless Assassin or an AIO. They both work. The choice is mostly down to aesthetics and cost - some people like the big dual tower look, others like the radiator. Some people will argue reliability in favor of big air, but If you don't go with a bottom of the barrel design, AIOs are pretty reliable these days.