“Ashkenaz doesn’t say Tefilas Tal.”
They absolutely do say Tefilas Tal. They say it before the private shmoneh esrei (amidah), thereby avoiding the awkwardness of the tzibbur saying something different than what the chazan says in chazaras hashatz. There are places that the tzibbur says morid hatal but the tefillah for tal is said afterward, in chazaras hashatz.
I do not know of any shul in Israel where they skip saying morid hatal after Pesach as in chutz la’aretz. Maybe there are a few, but I have never seen one.