Ozone only sanitizes the water that's in the chamber. Same with UV. If anything UV systems burn off the chlorine in the water going through them. Sadly we don't put much stock in those sorts of systems - a saltwater generator, on the other hand, keeps chlorine residuals in the pool.
Pool store testing is notoriously unreliable. They might be accurate but they might not - and we can't know which ones to trust. So we only trust test results taken by people and not by stores.
It's possible the Ozone/UV systems have kept your pool in check. Or that you have more chlorine or less CYA than you suspect. Or that you've just been lucky. But as bad as algae is, it's worse if someone gets a bacterial infection from the pool water - so it is really important to keep it properly sanitized.
Good plan. How do you currently chlorinate?
Oh, and if it's the K-2006 and not the K-2006C, then you'll probably be running out of FAS-DPD powder and CYA reagent pretty quickly. You might want to get some refills on those now too. See
Test Kits Compared for the number of each test you get. (The other ones are done so infrequently that you probably have enough in the K-2006...I don't think I'll ever use all of the TA, pH, or CH stuff that came with my K-2006C.)