Depends how you define cheaper. The visa sponsorship is a one time sunk cost, but if the candidate you sponsor aggress to work for several thousands or even tens of thousands less per year than the local candidates, then the saving for the company are obvious.
It's an opens secret that companie in Germany/EU usually sponsor the candidates who accept less than the market rate, otherwise they would just pay market rate or a bit higher and hire a local without the visa hassle. D'uh! Even my former German boss was open about this with me that the recruiting department is constantly trying to bring candidates from places with low wages and low QoL like Brasil (their example, not mine) in order to not have to pay higher than market wages to poach local talent.
>the minimum wage for a German work Visa is 56,800 EUR (EU Blue Card) and 44,304 EUR for shortage jobs.
44K/year seems pretty low for a so called "shortage job", and even more so if you consider the rampant inflation in the past couple of years. AFAIK 44K is pretty much around or sometimes even far less than a new grad makes in a skilled field at a good company. So, IMHO such low wages for supposedly "shortage jobs" are there for wage suppression and nothing else. If we had something like 70-80k/year as the bar for a shortage wage then we'd be talking sense.
The secondary effect of immigration is increasing the burden on inelastic services and assets that everyone needs such as doctors and rentals for everyone, as the real estate supply is highly restricted due to NIMBYism and several political and financial reasons around the rigged real-estate market I don't want to get in right now as that always makes my blood boil.