As the person who casts the first vote on a lot of "Speculative Answers" questions...
Is there something fundamentally wrong with just the sheer number of answers ... why not close all questions getting more than X answers?
There are legitimate, non-close-worthy reasons for a question to have a lot of answers. One of these is if there's a lot of bad answers: answers which are clearly and objectively worse than the intended one. Puzzling's solution to this, at least as far as I'm aware, is to downvote/delete the bad answers. I downvote usually, leave comments when I have the emotional energy*, etc. However I'm unable to vote to delete (<1k rep left!) and rarely flag**.
What if many are innovative?
If there's room in the question for innovation that is a problem. This is what's so hard about "soft" puzzles like lateral-thinking and odd-one-out: they have to be just broad enough for the answer to be fun, but not any broader or they invite too many answers.
I'd like to repeat the immortal words of our one and only @Doorknob:
The problem is answer validity, not quantity.
If a question has multiple valid answers, it is close-worthy. If a question has multiple answers, that's not necessarily close-worthy. The silhouette question you link has one answer which is clearly better and less arbitrary than the rest. In my opinion the other answers should really be deleted, or at least downvoted severely. However Amruth's puzzle had multiple valid answers and thus merited closing.
Please see Why are questions off-topic if they invite answers which are not demonstrably correct, or are otherwise speculative?
If you follow the logic of "many answers", then most Matchstick puzzles will not survive. Many "What am I" puzzles will not either. Riley's affix puzzle got 8 answers with probably more coming.
Yes. It is really hard to make a good puzzle of those types which will have a single objectively correct answer. Those are close-worthy if the other answers match the question as well as the intended one which is common because a lot of these puzzles are poor-quality/low-effort/non-well-enough-defined. If the others don't match well, again, time to downvote and delete. Personally I do the ones I see (though I don't see any affix riddles because I hid them from my homepage).
* In the case you link I did not have the energy to even downvote and was rather hoping someone else would do so. I still don't. I'm not sure why I'm spending so much energy writing this, come to think of it.
** I and the mods have some disagreements as to what is answer enough, so I've stopped trying to flag anything remotely borderline.
*** "Offensive" isn't a close reason, it's a flag reason. You are correct if you don't see any other triple asterisks in this answer, this refers to the question.