u/EbonRazorwit

There were rumors in canon about Stain being connected to the league of villains, and Class 1-A shuddered at the idea of him showing up at the USJ with the rest of the league, but if he actually was there, he would not tolerate the idea of Shigiraki killing All Might.

So, the second Shigiraki says they're going to kill All Might, he would try to kill him.
Assuming Stain was never told about their objective before they got there, and he was somehow convinced to come along in the first place.