Some of that's just how probability works. Like, let's say there were 100 artifacts, hypothetically. When you have three left to get, every time you get a random artifact you would have a 3% chance of getting one of the ones you need, so it would take an average of about 33 tries to get one of them. Then you have two left, which means with a 2% chance to get one of the two you need it would take an average of 50 more tries. When you have one left, it'll take an average of 100 more tries. So the last artifact will usually take about twice as long as the second-to-last artifact, and dramatically more than all your previous artifacts, just because you're looking for an exact specific one rather than one out of X you need.
And it actually isn't too unlikely for it to take significantly longer than expected average, too: for example, with 3.6% of getting any given artifact from the artifact trove (odds taken from the wiki) you would expect it to take about 28 tries to get Elvish Jewelry. But the odds of going 60 tries without getting it are about 11%: definitely bad luck, but "about one in every ten games" bad luck, rather than the astronomical odds it feels like.
Of course this is all assuming abstract "truly random" completely independent probability which probably isn't 100% being used given the existence of Stardew Predictors and whatnot? (I am a bit of a math nerd, rather than a programming one) But personally speaking I highly doubt Concerned Ape put in secret "choose a random artifact and make it deliberately harder to find just to mess with the players" code.
