yes, it is true that if you increase the supply of labour while reducing it's demand or otherwise holding it steady then you will suppress wages. that's economics 101. arguments against this rely on the question of what effect immigration has on demand, which is variable by sector. and, demand for labour is indeed rather low right now. but, wouldn't it make more sense to lift the lockdown? as demand is low for labour across the board right now, it's hard to see how banning immigration would have much of an effect on wages.
while i acknowledge that immigration is a factor in reducing wages in the southwest, i don't generally support policies intended to slow or ban migration into these regions. rather, i tend to point fingers at employers that hire people under the table, and a system that is erected to look the other way at it. so, i would rather see increased enforcement of the labour laws than bans on immigration. in the end, i would actually support a schengen-style agreement for the nafta countries that upholds free movement across the borders, but it has to be accompanied with stricter enforcement of labour laws, and the mexicans have to up their game on union rights.
if we actually enforced the labour laws properly, then mexican labour would be more expensive, and the issue would resolve itself.