Even the grots have their rebellions now and then
I think people are underrepresenting the Imperium, or at least overrepresenting the bad parts. Hiveworlds and life in the priesthood can suck, but thay are statistical minorities. Hives are the Adeptus Terra's version of Forge Worlds, so are planets dedicated to industry and thus everyone there works in the factory cities or to support them.
Most sources depicting "civilised" worlds (ie fairly anologous to modern Earth) show a world which whilst lacking in what we would consider basic freedoms such as democracy there is a great degree of self-determination (and many worlds do have elected leaders).
The concept of being born into a job in the Imperium is generally on aplicable to members of the priesthood or Mechanicus. There you have people who have hereditary roles as scribes for the Administratum or whathaveyou, but for many it's just a job. Ravenor has a bit where they infiltrate an administratum facility and there are people who are basically there as office workers. It's their day job, and it's not a great job but who here hasn't had a dreary job they hated? Ok, maybe it doesn't make your head spontaneously rupture, but still...
Obviously there are extremes which are more interesting to explore from a writer's perspective, but the Imperium has as much social mobility as our own world when taken as a whole - not just the 1st world - it seems less because the scale is so much bigger.
As an aside Space Marine, though no longer 100% canon, has a discussion on joining the guard. The main character lives on Necromunda, and is hoping to get into the PDF, so then he might get drafted for the guard and get off world. As you can guess from the title something else happens, but unless you're Cadian or a Krieger there's no guarantee you'll get into the guard - your planet might only be drafted once every hundred years for instance or be a planet that doesn't produce regiments for some reason (vital manufacturing / mining or somesuch).
Related thought: the 13th Black Crusade would have been dealt with a lot easier if they hadn't shipped approximately eleventy bajillion dudes off of Cadia across the galaxy. Hell, given how often Cadians turn up they probably could have kept them all there and launched a successful crusade into the eye