No. Basically look at the D2 schools in two groups.
There's 19 Division II schools with varsity men's hockey programs - 13 of them play Division I.
Alaska, Alaska-Anchorage, American International, Augustana (S.D.), Bemidji State, Bentley, Ferris State, Lake Superior State, Michigan Tech, Minnesota State, Minnesota-Duluth, Northern Michigan, St. Cloud State.
The remaining six are in the Northeast-10 conference and they play pretty much D3 schedules but aren't eligible for the D3 postseason. I am almost 100% certain they don't offer athletic aid for hockey (I'm actually checking on this for confirmation)
Assumption, Franklin Pierce, Post, Saint Michael's, Saint Anselm and Southern New Hampshire.
The college hockey fans would like the NE-10 schools to move to Division I to create an opportunity for a new conference up there. The women's varsity programs at those same schools compete for NCAA championships - since there's a "National Collegiate" Division (commonly, but not accurately called D1). They don't play D3 schedules like their men's programs do.
Oddly, two NE-10 schools - American International and Bentley - play D1 men's hockey.
There's also six Division III schools that play D1 men's hockey: Clarkson, Colorado College, RIT, RPI, St. Lawrence and Union.
Tennessee State, an HBCU with the backing of the NHL's Nashville Predators, has announce its starting a Division I program in 2025-26. They will compete as a club team this coming season.
The future for college wrestling isn't quite as similar. D2 and D3 are growing. D1 is trickling smaller.