If your bar for voting to have meaning is that your individual vote will change outcomes, you'll find that not a single model of democracy (much less implementation) is satisfying. That's an impossible, nonsensical bar. You can't possibly hope to swing a state to your party if everyone on the underdog side stays home because it seems unlikely to happen.
Your vote has meaning even if you weren't personally the arbiter of political change. Aggregates are defined by their constituent parts. You're acting as though voting is worthless if your cause doesn't receive enough votes to pass a cutoff point. It's true that it won't pass, but that doesn't mean the vote didn't matter. Apply this model of thinking elsewhere: if you made 2 points in a basketball game and your team lost, were those 2 points meaningless just because you didn't win? Should you recuse yourself from playing because it's unlikely your team can win? How about raising money for a Kickstarter - was it a waste of time just because you didn't raise the goal and won't receive the funds? Should you not try to raise money and find as many donors as you can, just because there's a minimum cutoff for payout?
Since analogies are imperfect it might be instructive to consider whether or not you can come up with a point at which you think your vote matters. If your vote is one of five, is that meaningless because two are on your side and three are not? How about one of 100? One of 10,000? One of 1,000,000? Can you - in a way that isn't arbitrary - come up with a point at which it stops mattering?
The fundamental problem is that if one vote doesn't matter, two cannot possibly matter either, and so on. Unless you disagree with very straightforward math, you can clearly see a single vote has some inarguable meaning. At that point you're just haggling on how much meaning it has as an individual, not on whether or not it has meaning. From there we can continue to have a discussion about why you should take this knowledge and vote with it, because regardless of your party affiliation the 2016 election was won with fewer than 100,000 votes. 1/n != 0.
On the other hand if you disagree with the basic math I probably can't convince you. But I can at least just tell you you're wrong without as much handwringing. There is no real paradox here. The only problem is that we as humans slip into intuition more than rationality. There is no perspective from which it is irrational to vote. It costs you nothing and can only have upside for your party affiliation. If you disagree we're separated by an axiomatic impasse, not a logical one.