Do miners validate each other's blocks? I understand that to mine a block, miners solve a cryptographic problem and the solution can be easily verified by other nodes on the network, appending the new block to their blockchain.

But why do miners consent that some other miner managed to mine a valid block before them?
Why don't they claim that some other miner's block is invalid even if it actually isn't, and buy time to mine one themselves?
Is it because all non-miners will validate anyway and miners don't want to risk having a discarded chain?
Mining is a random, progress-free process. Miners are essentially attempting to find a partial pre-image of a hash. Each new (valid) block candidate has exactly the same minuscule chance of yielding a new block. This means that finding a block for the next height is just as likely as finding a block that competes with the current chain-tip.
However, finding the next block is a lot more profitable: the current chain-tip has already been relayed to a large part of the network and all other miners are trying to find it's successor. Mining has a real cost, and only blocks that end up being a part of the best chain get paid out. To successfully compete with the current chain-tip, the miner would have to first find that competing block and then another to make it part of the best chain. It doesn't make sense for a single miner to try to find two blocks while all remaining miners together only need to find one more block to thwart him. Rather, the miner will spend a fraction of a second to validate the current chain-tip and try to find the next block.
There are two exception to the above behavior: Firstly, when a miner controls a significant portion of the total hashrate, they can theoretically attempt to increase their revenue disproportionately by starting to mine on new found blocks before publishing them, gambling on finding two blocks before the remaining network does. This attack is referred to asselfish-mining. Secondly, when a miner controls at least half of all mining power, they can maintain a monopoly on creating blocks by ignoring all foreign blocks and outhashing all other mining power in a so-called majority-attack. This is a popularly discussed failure scenario for Bitcoin and would likely undermine trust in Bitcoin altogether.