Actually, Old Faithful is almost 7500 ft. - one of the "lower" points in the park.
Much of the park is 7500-8500 ft. - so summer comes late and departs early.
One should expect snow and freezing temps in June.
Many park areas - campgrounds/concessions - do not open until late June.
Few services are available before Memorial Day - with the exception of Mammoth.

I have ridden early season - after roads are plowed but cars are still excluded.
One trip in the 1990s I was the only person - not one other all day.
Rode along Yellowstone Lake with giant icebergs on the shore.
(They no longer allow early riding along Yellowstone - grizzly habitat.)
But much of the riding was also just between walls of snow.