Here's an simple analogy,
Think of your coax as a garden hose, your radio is the faucet, and your antenna is the end of the garden hose.
A high SWR would be like putting your thumb over the water flow creating a backflow and the excess water spraying out all directions yet still much of the water will still shoot out straight.
The higher the SWR would be like putting your thumb tighter over the water flow creating more resistance making more pressure to force the current flow of water out.
Yet despite all this, all the water will flow out until the water is shut off just like all the RF current will shut off when you unkey the radio.
A high SWR will allow the flow of RF current out but a lot of the reflective RF will radiate out the antenna as unproductive heat just like the wasted excess water spray like a hose with your thumb on it.
This can cause a radio to push harder to compensate generating more heat in the final transistors which can lead to failure.
If RF current truly backflowed INTO the finals, then that would be catastrophic.
So no, reflected watts of RF power doesn't go back to a radio. An SWR meter can measure the amount a relective loss which can determine relative power out.