They don't have an expiry date. My tri-bike (which doesn't get a lot of miles) has a mismatched pair of tires (gatorskin on the front and some kind of specialized allcondition tire on the back). Those tires are at LEAST 10 years old. They're fine, no signs of delamination or cracking.
You're luck is better than mine, then. I just replaced all the family bike tires in my house because they were delaminating. All were somewhere around the 10 year old mark.
That said, you are probably right....sitting on a shelf (new), inside in climate controlled conditions waiting to be used, they be fine.
But I'm weird. I like my old aluminium p2k, my non-split-nosed saddle, my vision tech non-adjustable aerobars, my Swedish goggles. And I couldn't really give a crap what tires are on the bike for training, as long as they're hassle-free.
Hey! I'm just as glacial when it comes to bikes. My TT bike is the same vintage as yours (p3sl v p2k). And my road bike is another 10 years older than that! I still have the original 8-speed Ultegra-6400 on it, too.
I'm not that picky about training tires. I tend to buy Conti Ultrasport II, just because I've had good success with it and its predecessors over the last 30 years. But, I also use up my race-tires after they have seen enough race miles that I want a fresh set for the next season.
So, between those two factors I just don't know that I would EVER go through 5 sets of tires (by myself). I've been trying to add up how many new sets I've bought since I stopped being a pure cyclist---I'm not coming up with a big number.
Also, I don't like to just store crap forever. My house is full of enough other shit (which I'm always trying to get rid of), I don't need to add to the mess with a lifetime's supply of cheap-ass tires.