Generally, the deer are only overpopulating (at least to the extent you describe) in areas that aren't hunted, which also tend to be highly populated. It seems many people enjoy seeing the deer in their backyards and don't want their almost-pets to be killed, especially if it means it might have to be done on their land. How do you propose dealing with that opposition?
You could also get more ppl into gardening. Gardeners end up hating deer after their gardens get raided once or twice.
Central Massachusetts has a very active deer hunting community. I never have any trouble filling my freezer with free venison every fall. But the deer population is still four times the sustainable level as set by the state Wildlife dept and local conservation groups.
Hunting isn't necessarily enough to control the population.
If your area doesn't have the access issue, then I would guess it could be an issue with the number of hunters - if there aren't enough, then they can't harvest the number required to bring down the population (or the limits make it difficult - seems like that's the case the way they handle doe permits). Many states have a program that allows hunters to donate their deer to a food bank through a participating butcher. This can make a big impact in areas that allow more harvests (MD allows 10 doe per year/season without any special permits).
White-footed mice are the most common host for the small mammal portion (but there are many options), and deer are the most common host for the large mammal portion.
Deer are easier to count and control than mice/voles/moles/chipmunks/etc.
So controlling deer populations is more likely to be successful in breaking the deer tick lifecycle.
So sayeth a publication from researchers at the University of Connecticut, at least.
I'm sympathetic with the almost-pet Bambi-loving crowd. But Lyme is real and not at all cuddly. Bring back the wolves!
The advantage of going after the mice is that they can use crispr to give them immunity. It could be used on deer, but would probably have more push back. Tick tubes have already been shown to be effective at the small mammal level.
It will likely require a combination to control.
Interestingly, the coyotes in the US East are apparently hybrid "coywolves". They are larger than western coyotes, but still not big enough to take down an adult deer, and they still do not hunt in packs.
If you have a large area to cover, Tick Tubes(tm) can get expensive. You can make your own pretty easily for about 1/20th of the price: toilet paper rolls, cotton balls, and Permethrin diluted to ~3%.
"...but still not big enough to take down an adult deer, and they still do not hunt in packs."
Coyotes will hunt in packs or alone. They generally go for fawns, but they do occasionally take adult deer (there's even video of it).
https://www.pgc.pa.gov/Education/WildlifeNotesIndex/Pages/EC...
https://www.trappermag.com/article-index/deer-predation-coyo...
Wolves are bigger, stronger, and cooperative in larger packs. Coyotes are more opportunistic and ad hoc, and easily scared off.
The net is that wolves have far more success with large prey.
Feral (or wandering) domesticated dogs will sometimes hunt together too, but their coordination is limited. Wolves are impressive creatures.