I can see a couple of problems...

Male ticks still bite, so they'd still cause problems. Although presumably lab ticks would be disease free.

A bigger issue is spread. Mosquitoes fly so you can do a release and they'll spread over a large area. Ticks walk. They can't travel as far. I suppose you could drop them from drones or a plane.

They'd cause problems for a few generations, until their population collapsed from a lack of females.

I never thought 'Jurassic Park' would become a prescient warning.

Should we? What is normally eating the ticks or living off of them?

You also have to ask the other way around: What do ticks do to their hosts? Animals such as moose or reindeer can suffer greatly from heavy tick infestations. Depending on the death rate this causes removing the ticks could have similar effects to removing predators such as wolves from the ecosystem. However there are many tick species, and I doubt the impact would be that strong, but who knows.

Certain ant species can eat ticks, but as far as I'm aware there's no species feeding only off ticks, and Lyme and Encephalitis are really nasty. But it's an interesting question: suppose we can somehow erradicate all ticks, what effect would that have? How much do they matter for the food chain or on ecosystems in general?

Ground birds as well. Chicken's (and Guinea Fowl) will eat large amounts of ticks if present.

It is a part of the possum diet. They eat upwards of 5k per year.