I think that a complete cooling solution with a heatpiped heatsink and a fan bracket could probably be made. Every single x570 uses the same size chipset, it's only down to the mounting holes and clearances.
Something similar to what Thermalright used to make would likely work if clearance is alright.
Clearance would be like so:
(Image from https://www.extremeoverclocking.com/...SLI_IFX_4.html
alternative installation orientation:
This is especially a good proposition for less expensive X570 boards which likely use cheaper fans but might have half decent VRM for the CPU. At 15W you could probably use a CPU heatsink designed for slim 15W ultrabook CPUs as they typically have a heatpipe to transfer heat elsewhere. 6mm heatpipe can already transport at least 40W and 8mm can do 60W , those are the two options below 11mm height of the fan from the Taichi (which would allow for <15mm with mounting hardware). This would solve the clearance issue for high end GPUs which are typically over 9.5" long these days. Obviously this voids warranty.
Looking at 11mm tall copper heatsinks you have Enzotech's CNB-S1L http://www.enzotechnology.com/cnb_s1l.htm
For a 15W chip if it's around 5°C/W for the Enzotech heatsink without even accounting for thermal resistance of the chipset itself and the thermal pad in between, it's going to be hitting over 105°C which is likely going to fry. Obviously a heatpiped solution (to avoid the clearance problem) with more fin area is needed.
For example the X570 Aorus Elite might be worth changing in this fashion since chipset heatsink is not integrated into the M.2 heatsink setup. This means the whole chipset heatsink can be removed without affecting the M.2 slots. Something built with the M.2 shroud as part of the chipset cooling solution as the MSI Ace, Taichi or Steel Legend are not conducive to any modification. Lower end boards such as the MSI Gaming Plus simply aren't worth thinking about since the VRM just adds doublers over the B450 versions ; the MSI Pro Carbon and Edge have the chipset cooling a part of a M.2 shroud as well.
Clearance is at least 4x2" with 10-15mm Z height on the Aorus Elite. It might be closer to 4 x 2.5" (metric I would presume ~100x60mm).
Obviously upgrading your cheap fan with another cheap fan isn't going to solve anything. I think the best option is to have a 90mm+ fluid dynamic bearing (i.e not sleeve or ball bearing) fan permanently on the chipset at 1000-1200RPM except with cross-flow rather than radial blower style. That way the air doesn't get recycled and you retain front to back airflow in all normal orientation ATX cases.
P35 was a 16W TDP chipset too. People are losing their minds over a 16W TDP... when it's been done easily without using a fan even if people don't find heatpiped heatsinks attractive in their RGB light boxes.