Yes, the only reason ASEL turbine time has historically counted towards major airline minimums is because some fighters only have one engine. Just because they hire harrier and viper pilots doesn't mean they'll extend the same welcome to PC-12, etc experience.

In this climate anything's possible, but it's hard to predict who might call and who might not, and how much total experience you might need. So something of an uncertain path. I think you'd get called eventually, but not necessarily by your first choice.