Yes and that's always the discussion when people from US come in to visit to the french part of companies. "Hey we heard that you are paid 5 to 10 times less than us, why do you accept that?"
Trust me, it's worth it. Really, it's not that much effort and it's worth it to signal to recruiters your real worth. And don't forget not all recruiters are equal.
And the visa issue is way overblown. It's easier than you think for a real engineer with experience.