Makes a big difference. Applying for work visas for someone from the EU is totally different than applying for a work visa for someone from the middle east.
I'm not an EU citizen. I doubt top tech companies would reject a candidate just because it's more visa work. They usually have an entire immigration department just for handling these cases.
You have no idea what a pain the US visa process is. For instance, for regular employees your employer needs to obtain a H1B visa. There is a maximum of 85,000 a year, which is reached in one week for the last 3 years.
Did you read what I said? I've went through the visa process.