The best way to remove the metal backings is to remove the dash and un-twist the metal tabs/flanges securing the plates.

Another way, which requires finesse and a steady hand, is to carefully grind the metal plate using a Dremel tool and metal bit at the edges where the tab goes through the vinyl. The cut off tabs will either push through using a small screwdriver/pick or can be left in place.

Go slow and gentle. You want to keep the heat at a minimum to avoid melting the vinyl.

How does the thickness of the replacement pieces compare to the original wooden strips?

If they are the same thickness they are designed to be glued to the metal plate and not to the vinyl. Use an epoxy and try to avoid getting the epoxy on the vinyl as much as possible.

Do NOT use "Gorilla Glue" or its ilk. Those styles of glue expand as they cure and usually either ooze out around the edges or bubble the wood.

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“Whatever story you're telling, it will be more interesting if, at the end you add, "and then everything burst into flames.”
― Brian P. Cleary, You Oughta Know By Now