I think it's totally unacceptable that regular people should have to "beg" the state for permission to go abroad!
The state belongs, to the people --- not the other way around! Who did they think they were, deciding who could or couldn't travel abroad? It's outrageous! I can see people really got fed up and angry about that.

Some Soviet people definitely went abroad though; a ferry from Leningrad docked quite near my house when I was a kid, and there were groups coming off that ferry. I always had the impression that they were travelling for work though, like people attending a conference or something like that. Never any children, only adults. Plus there were plenty of Soviet lorries arriving on the ferry - could be recognised because of the Cyrillic letters and unusual makes. It was not unusual to see Soviet people (Russians and Estonians) in Helsinki. They stood out a bit; different style of dressing (and ugly glasses!) I used the Helsinki - Leningrad train with my choir when we went to a choir festival and most of the people on the train were Russians, including families and lots of old people. Finland didn't allow any Soviet people to stay (defect) though.

I remember being told that Eastern European states at the time prevented people from leaving because they didn't want to lose qualified people whose education they had invested in... But that seemed false somehow, becuase they were not the only country offering free education to citizens!

But USSR sent thousands of Soviet citizens (Estonians) to Sweden. They had said they wanted to emigrate to Sweden.
Later Sweden found out why the USSR didn't mind losing them: They were criminals - some really nasty and violent ones; ended up running a mafia.

Plus LOTS of Russian Jews left for Israel, didn't they?