That is reward for proof of work provides incentives for miners to validate transactions; it does not validate them through proof of work.
You are correct in that the proof of work itself does not validate transactions, that is done separately, so that only valid transactions will be included in the block.
However, the block itself must contain a valid proof of work (the block hash). Without this proof of work, the block would not be valid (even if the transactions are valid). So it can be said that a block is made valid by its proof of work.
Edit: it is worth noting the distinction between the act of performing the proof of work hashes, and the object that is the proof of work (block hash). My answer references the object, not the act.
"Validation through proof of work" is indeed a wrong description of what mining is. Mining is what operates a universal clock of sorts which solves the problem of verifiably ordering the blocks of the blockchain. The Bitcoin paper refers to it as the "timestamp server".
Note that miners do not verify blocks. They get a very rare and expensive chance (once every 10 minutes on average, enforced by Proof-of-Work) to propose a block, which better be valid or they've wasted their electricity. But it is all other full nodes (including the ones that are not mining) that validate the block proposed by the miner who solved it and broadcast it to the network. The nodes do this by simply ignoring it if it is not valid in their "opinion".
For a much more detailed description of what I mean by "timestamp server" or "clock", see this write up: https://grisha.org/blog/2018/01/23/explaining-proof-of-work/