You're referring (I think) to Arrow's impossibility theorem, and "voting can't be fair" is an extremely common misinterpretation. What it actually says is that no rank-order voting system can simultaneously satisfy all of Arrow's criteria for the ideal voting system.
First of all, the theorem only applies to rank-order voting, which is what most people are used to but not the only option out there. Rank-order voting is when you sort people in comparison to one another. FPTP is an example of rank-order voting, although the only ranks are "#1" and "everyone else". Another well-known system is the Borda count, where you rank everyone on the ballot, 1 through N.
However, there's a totally separate class of voting called rated voting, where you judge each person individually. For example, on this site you don't sort the posts from best to worst; you take each one and either upvote it, downvote it, or abstain. This is essentially range voting (with the range set between -1 and 1), which is a type of rated voting. Arrow's impossibility theorem says nothing about rated voting, so it's possible for a voting system in this category to be fair, and indeed range voting has many advocates (I won't go so far as to say it's "best" because I'm not sure the experts will ever agree on that).
Besides all that, Arrow's original criteria are quite strict. In particular, independence of irrelevant alternatives (adding a new candidate to an election shouldn't change the result unless they win -- it shouldn't result in another candidate that was in the election suddenly stealing the win from the original winner) is difficult to fully satisfy. Unfortunately, without it the system tends to suffer from the problem you mentioned upfront, strategic voting. For example, the Borda count is a rank-order voting system that satisfies all of Arrow's conditions except IIA, but is almost hilariously vulnerable to strategic voting -- you always want to rank your candidate's strongest opponent last, even if you actually like them second best.
As to the last part of your question, using a weak voting system is definitely a real threat to fair voting, but Arrow's theorem doesn't just boil down to "democracy is impossible". There are many voting systems that are quite resistant to strategic voting, vote splitting, etc. The real problem is that we tend to not use them because they require more work. Plurality voting is attractive in that it requires checking one box, so it ends up in common use despite its problems. In short, we shouldn't worry too much about whether or not a voting system is perfect; picking one that's pretty good would be a vast improvement over the current situation in most elections