In the US, most H1B hires I personally know have had their green card sponsorship started within an year of hire. Then it takes about 2-3 years to get it finalized. I knew of one who even got the process started in 2 months.
So there are some differences from other countries but no 5-10 years wait.
Wrong. The green card on EB2 takes 6-8 years for Indian and Chinese born immigrants. On EB3, it can take over 10 years. The green card process involves a per-country quota. Folks from countries other than India, China, and a couple of others get it "finalized" in 2-3 years.
I mean, having been in the country 5-10 years, depending on the country, is a requirement of getting the citizenship, not just to to get the process started.
So there are some differences from other countries but no 5-10 years wait.