I know tons of people (including family) who are/were here on h1s of various sorts, artists visas j series visas and people who've come in through the whole apply for visa come over blind sort of way.
And frankly for the work related visas you already have to have the job. You might be hired specifically for a visa job but job comes first. So too student and research visas. Already accepted and enrolled, coming over for a specific fixed research project. Leave school, leave job, lose visa. There's your "connections" right there. Artist visas require you to already be successful and notable as well as have a plausible reason for working in the US. If you're not you have to come in on a sub visa connected to someone who is. And you have to show that you're an essential part of their work.
In all cases you need a sponsor or something similar, especially once it comes to applying for the greencard. There are regular trips home before getting a green card in most cases to renew the visa. Once you apply for the green card you cant leave the US for any reason or you have to start over fresh. Everyone I know keeps an immigration lawyer on retainer, even if its provided by an employer or school. The one, one person I know who's successfully come over through the visa lottery system and navigated it to citizenship only got his visa in the first place because he served with a US marine platoon as part of an Italian military unit. Just qualifying to apply for a green card can take a decade. Getting through that process is multiple years, with no guarantee of success. Getting citizenship after that is also many years.
Frankly it's not something you can or will be able to do without money and people in the US. It's next to impossible to do it right otherwise. And in any situation you spend a decade or more in a legal limbo where one mis-filed piece of paperwork, one trip home for a funeral, one financial set back sees you deported or starting over from scratch. It's insane. And it's relatively new. When my grandfather came here in the 50s it wasn't nearly as complex or time consuming.