Why do miners validate transactions?
Because they want to create blocks, that contain valid transactions. After sending the block to the network, other nodes verify the block for correctness. If the block is rejected, the possibility to "win" is lost...
you seem to postulate, that there are many invalid transactions. I am not convinced, this is the case. Yes, everyone can send a tx to the network. This tx would end up in any of the approx. 10.000 nodes. There are far less miners, so the full nodes already filter invalid transactions. A particular wallet is usually not connected directly to a miner. As full nodes validate transactions, the probability that a full node forwards a wrong tx is minimal. If a full node does, there is other nodes, which verify the tx as well, and if this nodes sends faulty tx, he gets ignored by other nodes. I just wanted to state this, to make clear, that malicious crafted transactions have a hard time to make it through to a miner. And yes, the miner himself can generate false transactions, but then again looses all possibilities to win a block.
Is the reason that there are enough invalid transactions to make this
an ineffective strategy, because the risk of having mined an invalid
block is not worth the time savings?
As I tried to explain before, invalid tx are not a common thing. Miners strategy is currently to win a block, maybe with some fees along. So each miner can decide individually, what he puts as "winning strategy". There are often blocks with zero transactions, or very view tx, cause currently the block reward is much higher than fees. This seems to be a valid strategy, not the invalid tx.