Replace a UPS battery or buy a new UPS?
I've never heard about replacing a battery causing the UPS to never be the same. The only UPS units that I've ever seen are just sealed lead acid batteries that wouldn't cause issues with the UPS electronics unless they weren't specced right (proper voltage and amperage rating). I've used APC and third party batteries and had luck with both. The third party batteries are generally cheaper so that is the way I would go. You don't need to replace the UPS unless there is something inherently wrong with the UPS electronics.
I would say that it's definately worth buying a replacement battery. Not to mention is probably better on our environment if you're only changing the battery and not the entire unit.
We've replaced the batteries in our oldest UPSes 3 or 4 times, and I don't see any reason not to keep them going for a while longer. These are a 5U form factor, 3000VA, 120V input and since it doesn't look like they're available any more, I hope they keep running for a while.
We have about 8 of the smaller 1200-1400 VA SmartUPSes and over the years the electronics on 2 of them have failed, but other than that we've replaced the batteries in those a few times as well.
I've bought 3rd party SLA (sealed lead acid batteries) for my electric bike, but it never occurred to me to look for UPS batteries from those suppliers. Next time we need replacements, I'll look into it. As far as I know, there aren't a lot of secret design techniques for SLA batteries, so I'd expect batteries with the same ratings to be just as good no matter where you buy them from.