Edit: I am taking a break now and will return later this afternoon/evening to respond to any comments and answer any questions. Thank you everyone for a great and engaged AMA so far.
* By law, the US can only issue 140,000 employment-based green cards per year, and no more than 7% to one country. This means people from India or China can face a 100+ year backlog, even after they have proved they qualify for a green card. There's no cap on marriage-based green cards.
* Processing times for many green cards (i.e. for people who have already qualified, but just need the physical green card), are 12-24 months.
* USCIS still expects many applications to be sent by mail. Some applications (like O-1s, EB-1s) require hundreds of pages of evidence, and it all needs to be printed out on 8.5x11" paper, for USCIS to scan it in on B+W scanners. This means that there is no error checking (e.g. on fee amounts), and if you have made a mistake, you might not know about it for weeks. Also, it means your petition cannot include working hyperlinks, webpages, or videos - the USCIS officer judges the petition by scrolling through a 400+ page PDF.
* The 'standard' post-graduate work visa is the H-1B. It's entirely lottery-based, not merit-based, and typically there are 400,000+ people competing for 85,000 visas. Many qualified people are forced to leave the US each year because they didn't get selected in the lottery.
No other country in the world requires foreigners to be significantly more qualified than its own population. You can move to France with a regular paying job no problem or just a few thousand euros in savings. Impossible in the US. You have to be extraordinary (they literally call their criteria, "extraordinary abilities") or you have to make top 5% money (so if you work in tech, that would be at least $500k-1M/year in many cases).
The only other way is to get married. This means there is a massive discrepancy between the qualifications of self made immigrants, versus those simply lucky enough to fall in love. It's pretty unfair, but that's how it works. But that's also the reason so many immigrants are so successful in the US, the bar is so high, that it creates a massive motivation to succeed to become eligible for the criteria.
The path is H1-B -> Green Card -> US citizen (I have done it), and to get H1-B your potential employer gotta post that $60-80k/year job and show that there were no qualifying US applicants for it.
Unless something has changed dramatically in the last decade this is patently false. Getting an EU work permit was historically very hard with employers having to demonstrate that a position can't be filled by an EU citizen before a non-EU citizen candidate can be considered.
There is no instant green card.
If a truly extraordinary Indian-born person (say a Nobel laureate or olympic gold medalist) files for a green card today, they will be waiting for 7-10 years to get a green card. At this point, it may be worse too, because this category's priority date has not moved a single day in 8 months.
In most European countries, once you're in, you can find a way to stay. One exception I can think of is Switzerland, which can be pretty annoying for temporary visas because they don't count for time accrual.
Austria has a pretty good system (RWR) that lets you job seek and is a pathway to permanent residency as a 3rd country citizen. I think there are similar programs in France and Germany.
For example "very highly qualified" in Austria is satisfied by almost anyone with a STEM degree, being under 35 and (amazingly) being an English speaker. If you have that initial visa, companies can hire you without worrying about sponsorship.
You could also use that as a route to the Blue card I think. I wouldn't say the bar is exactly low, but a lot of mobile people are sufficiently educated and are paid enough. As in, a typical European STEM salary would cover it.
But also the grandparent's comment is out of touch. Of course countries want people who are more skilled than local labor, that's the whole point. Aside from the benefit of attracting talent and higher tax revenue, it's much harder for your voters to argue that immigrants are taking your jobs this way.