Looking at it solely from a perspective of competition between labor glosses over the fact that insufficient labor is also bad for the economy because it keeps companies from growing and hiring more people.

So sure, while the fewer jobs that they can fill could have higher wages (not a given, because lack of labor can stunt or kill companies) there could be much fewer people employed overall, which is clearly bad overall.

Of course, that assumes there is enough room for companies to grow. There are strong indications (e.g. the various labor and unemployment surveys) that this is the case in the US. In fact, there is a credible theory that the reason the US managed the inflation crisis so well was due to the immigration crisis.

I elaborated more (along with a couple of relevant studies) here: https://news.ycombinator.com/item?id=45308311