Canada

most anti-immigration arguments completely forget that they help with the supply by bringing manpower to build the homes.

To a degree. Canada's point system for immigration selects white collar workers, while most students coming in on education visas are aiming for office or healthcare jobs. The last time I looked, the trades had the same proportion of new Canadians as the rest of the population - so yeah, some immigrants are in the trades, but they aren't overrepresented.

The federal government should be selecting for more construction workers. IIRC there are a couple of programs that encourage tradespeople to immigrate, but they don't bring in a significant number of people.

Bringing more manpower doesn’t help since it’s not the bottleneck.

Lack of construction workers is a bottleneck. It's just one of many, including: increased construction costs, developer incentives, zoning, lack of government construction, etc. Like you say, we need to solve all of those at once.

The anti-immigration arguments are mainly from the racist right that looks for a boogyman while ignoring the real causes of our issues.

I don't think that's fair to say. Immigration exacerbates the housing/healthcare crises, even if it isn't the sole cause. Lining it up on political lines turns immigration into a wedge issue, which doesn't help anyone.

It's much faster for the federal government to reduce the number of newcomers than it is to build houses, train healthcare/construction workers, etc. A reduction would reduce strain on our society while we fix the many problems we're facing. Once housing is again affordable, and every Canadian has access to appropriate healthcare, then we can see about increasing admission if appropriate. In the meantime, we really need to increase the number of tradespeople (and healthcare workers) we're bringing in (and certifying) even as overall numbers fall.