This probably wouldn't help much, and might hurt
The purpose of mandatory tags is, at least partly, to keep out offtopic posts. Now, it's true that a post that doesn't have a win condition (and isn't tips) is offtopic on PPCG. However, when an offtopic question is posted here, it's often clear that the question asker struggled to find even one currently existing tag to tag the post with, and it's often tagged with a nonsensical set of tags (e.g. internally contradictory, and also failing to describe the question). (programming-puzzle seems popular for that; "my question is about programming, and I'm puzzled"!)
It's also the case that a) a question which is very offtopic is trivially identified as such and tends to be closed within a few minutes, thus doing no real lasting harm (we have a lot of moderation power within the general user community on PPCG compared to the amount we actually need, so wasting small amounts of moderation power isn't an issue); and b) a question which is only slightly offtopic nearly always has a win condition tag, so this wouldn't affect it. The only occasions I can really see this helping are situations where a poster simply forgot to write "shorter (measured in bytes) is better" in the question (or whatever the win condition they wanted is, but code-golf is the most frequently forgotten), and the "you must provide a win condition tag" prompt reminds them. That's rare (although not unheard of), and commenters to the question are likely to remind them quickly enough.
I can see two ways in which this could hurt, though. One is giving people who post wildly offtopic things a nice handy list of tags that they can use to get their question past the "this is offtopic" check; it's not like we have a classification as simple as discussion, support, feature-request, bug that clearly delineates the site into only a few possible valid reasons for making a post. (In fact, we even have code-challenge, which is a tag which explicitly means "none of the above" when it comes to win conditions, and thus any possible post is covered by some win condition tag, even if it doesn't have one in its body; this is directly counter to the main advantage from required tags.)
The other issue is that sometimes, a win condition becomes popular enough to be worth its own tag. The OP doesn't list cops-and-robbers or answer-chaining, presumably because they weren't around at the time; we've had to add them since. If we needed to bother a diamond moderator (or more likely, SE staff) to adjust the list every time a new win condition became popular, the list would likely be doing more harm than good as it would cause a limbo situation where we couldn't use the new win condition tag until a sufficiently highly-privileged user was found to update the site configuration.