When the actual power used by the CPU reaches 350 W, that happens because the BIOS of the motherboard configures the CPU for an unlimited power consumption.
In this case the CPU consumes as much it can before the temperature exceeds the temperature limit. The actual power that is consumed will be determined by how good is the used cooler.
The MB manufacturers do this so that their motherboards will win the benchmarks in published product reviews.
Unless the BIOS locks the power control registers, you can configure any Intel or AMD CPU for any power limit you desire and the power consumption will never exceed the set limit.
A low power limit will diminish the achievable multithreaded performance, but unless the limit is very low the multithreaded performance per watt will be greatly improved.
The TDP matters a lot for a laptop or for a SFF computer. In this case the laptop or small computer is really designed for a definite TDP, e.g. for 28 W or for 45 W, and attempting to make the CPU consume more than that will result in thermal throttling.
For a desktop, the design of a motherboard does not depend on the TDP, because it does not include the cooling system. The MB designer only takes care that the voltage regulator should provide enough current to reach the highest power consumptions that will not destroy the CPU.
The actual power consumption that can be reached without thermal throttling is determined by the case, the case fans and whatever CPU cooler is used. For most desktops it is easy to ensure that the CPU can consume more than its nominal TDP, without overheating.