In my area of eastern Long Island, we did not have serious problems with deer ticks and Lyme disease for 40 years before deer arrived in the early 1990s and the deer population exploded.
Once the latter occurred, deer ticks and Lyme disease became major issues. Lone star ticks and a variety of other, serious tick-borne diseases have since been added to the mix.
As an experienced professional mammalogist, I know that smaller mammals were abundant from the early 1950s to the early 1990s but unable by themselves to maintain rich tick populations (other than a moderate abundance of dog ticks).
Japanese barberry and honeysuckle may promote tick abundance in some areas; however, the former has never been a significant part of our flora, while honeysuckle tends to be patchy and dispersed in its distribution. In the presence of numerous deer, other vegetation and forest litter seem fully capable of supporting a superabundance of ticks.
Those who deny a connection between the presence of too many deer, an unacceptable incidence of dangerous tick-borne diseases, and widespread natural habitat destruction (with a concomitant loss of biodiversity) in many areas are doing us all a serious disservice.