While I understand the basic concept of having Fun on the Puzzling site, I wonder if the First to Answer importance promotes more viewers or discourages some. Here is my issue. This is a global site. So if someone posts a question when it is night in US, the US guys (in the morning) most often see that the question is already answered. It is possible that the "late viewers" might actually solve the puzzle in a shorter time than the the first to see/solve person but they either don't attempt it or ignore it. Of course this is a speculation. There may not be a solution to this, but is it possible to put a timer on the puzzle whereby once you open it the timer starts and it stops after you post the answer? This may require "honesty" on part of the solvers so may be very tough to implement. Comments are welcome.
This is a symptom of a more general problem, which is that the Stack Exchange format is not a good match for puzzling. There are several other inconsistencies, into which I won't go now, but the basic one is this:
PSE looks like all the other SE sites, works like all the other SE sites, and in every possible respect, it is exactly like all the other SE sites. Except one: You are not supposed to use PSE like all the other SE sites.
Being the first to answer or the one to get the tick is of course satisfying and gets you the yummy peer validation in the form of "rep", but to paraphrase one of my favourite shows: on PSE the questions are made up, and the points don't matter. (Answer upvotes being worth twice as much as question upvotes should make that exceedingly clear.)
Instead, to best enjoy PSE, you should forget everything you know about Stack Exchange, go look at the questions with green in the score box (gasp!), read the third highest voted answer (it may have a brilliant observation in it!), try to find answers to already solved questions, and in general, do anything you would do on a puzzle site expressly designed for the purpose.
Or to really boil it down: The only important part of PSE is the P.
And finally, to answer the question: It is perfectly ok to not look at the earlier answers. The puzzle is just as good even if you don't get fake internet points for solving it. We should advertise this approach to all newcomers, (at least on The Tour) since they have probably visited some other SE sites earlier, and will most likely feel conflicted about doing such things.
As a relative newcomer, and a US guy to boot, I'm strongly encouraged to continue visiting PSE as it stands. When there's a great answer, accepted or not, there's usually some wiggle room to - as you put it - have fun. Or to learn something. And I have seen many collaborative, partial answers, especially for the complicated ones, which is just fantastic to watch. Plus there are always the weeks/months/years old unanswered questions (like the ANALOG contest). Not sure a timer would do anything but discourage thinking about it, on your own time. Maybe some of us have more time than others, but they're likely the ones who are going to beat the timer anyway. I'd rather everyone think about it at their leisure, and give us their best.