Yeah the full answer is “it’s complicated and yes maybe some people see wages depressed or increases that would have happened, slowed, by immigration”. It can, for a given individual or even sector, do the thing people are worried about, even if most benefit—mean or even median wages tending to go up isn’t the same as your wages will go up. Simplified “it doesn’t lower wages” messaging has a smell to people burned by other neoliberal policies, and they’re not wrong to detect a hint of the ol’ BS, even if their concern is overblown or misplaced.