It's very hard to answer your question because this question is very, very opinionated, if you will. Also, if by Russia you mean the Russian Federation (or, RSFSR if we're talking about Soviet times), this republic was/is multinational, there are a lot of other ethnicities there, and some of them are quite large: there are 5 millions Tatars in Russia now, 3 millions Ukrainians and so on (and I believe that real numbers are even bigger, because some people would call themselves Russians, despite their origins).

So, Russia isn't for Russians only (and never was).

Even the main Soviet rulers weren't ethnically Russians sometimes. Stalin isn't Russian at all. Brezhnev was born in Ukraine. Lazar Kaganovich was Jewish and so on.

Regarding the second part of your question (how all these people were content to be part of the Union), all we can say it varies. I just think it needs to be mentioned that almost all of these countries weren't independent before the USSR, they were parts of the Russian Empire. So, their independence wasn't completely out of the question, but, I believe, it was a really extreme view (and, I think, it was a criminal offense to even discuss it, so we don't have any real data about how popular these ideas were). Most people, I think, didn't care about this. There weren't living Armenians who remembered independent Armenia, for example.

The one obvious exception is Baltic countries, because these countries were independent in 1918-1940. There were a lot of people who remembered independence of their own countries.

Also, I don't think there was a real Russia vs other republics antagonism. There was Moscow vs other republics antagonism. It needs to be said, that Moscow understood this problem, and tried to solve it by using locals to manage their own republics. Actually, I believe at the start there was an idea to eliminate ethnicities as a factor at all (including the Russian one) and to build the new Soviet character, but it failed, and Moscow rulers loved to install locals as heads of the national republics. There were several attempts to ignore this practice, but IIRC none of them were successful. So, it means, that Georgia was ruled by Georgians, Ukraine was ruled by Ukrainians, Kazakhstan was ruled by Kazakhs and so on.

Of course, they weren't independent in their solutions. Their masters were in Moscow. Some of them were Russians, and some of them weren't, but I don't think this factor was really relevant. Their Communist ideology was much more important than their ethnicities.

What I'm trying to say that this discourse—Russians vs Locals—wasn't the main one at those times. There was Communist vs NonCommunist antagonism, and Moscow vs all the others.

(And I have to say, that Moscow vs all the others is still a thing in Russia, but within its own borders).

After the collapse of the Soviet Union, independent republics had to invent their own independence from scratch. Most of them choose to build the national state. And the national state requires the national idea. And the national idea often requires an enemy. 'Requires' may be a strong word here, but one of the simplest way to build your own state is to say what you want to avoid/suppress instead of what you want to build/support.

Russia was seemed (and rightfully so) as a successor to the USSR, so all the USSR sins slowly became Russian sins. Also, Russian Empire and the USSR after that wanted to make Russian the main language within its borders, so local languages weren't supported as well as they should, and this point became one of the main points for the post-Soviet separation. All these countries tried to restore their language as a state language, and many of them, doing that, suppressed the usage of Russian language in its borders. Russia was often portrayed as invader, although some of these invasions were made hundreds years ago, some of theme weren't real invasions, and some of them were made not by Russia, but by Soviet Union. It didn't help that Russia was perceived as influential and sometimes dangerous neighbor. It didn't help that there were a lot of ethnic Russians in some of these republics (and new national states were afraid that these Russians would want to become a part of Russia, with all these lands they lived in). Or, even worse, these people with another ethnicity weren't Russians, they were someone else (and there were/are several blood conflicts on the post-Soviet space because of that).

Soviet Union was relatively careless about its inner borders, and sometimes there were a contradictory solutions when areas with some ethnicities belonged to republics with other main ethnicity. It wasn't really important when Soviet Union was here, but it became a reason for wars after its collapse.

I'm not trying to say if these new national ideas were right or wrong. In some sense, they were inevitable. What else these new countries could do? I just want to say that Russia with its 'vassals' is a relatively modern narrative. It's quite convenient for these new countries that were created from Soviet republics. But, arguably, it's not the best one if you want to understand what were the relationships between Soviet republics and the center. It was slightly more complicated and messy and it was based on another set of principles and rules.