https://www.cdc.gov/lyme/data-research/facts-stats/lyme-dise...
That said, whether it is hotter or cooler doesn't make much of a difference in terms of how you go about your day - you pretty much have to assume you can encounter them regardless.
Up and down cycles in temperature have always been a thing on the North American continent but climate change has made it even more variable. We will still get places where it gets very very cold but not for the consistent chunks of time it takes to set back tick populations significantly.
TLDR I don't think it's the heat or cold per se but the variance.
And yes climate change is absolutely the prime factor in their spread. Into places where they were not ever a threat before.
It impacts the population, but even a couple solid weeks of -20C weather doesn't seem to be enough to eradicate them.