I'd think it was both points you noted. First, the 15 constituent republics of the USSR (also called "union republics," the top level of administration) all did become independent. They are the modern countries of the post-Soviet Union today (Russia, Ukraine, Belarus, Kazakhstan, etc).

The second highest level of administration was autonomous republics. The vast majority were within what is now Russia (think Chechnya, Sakha, Dagestan, etc; there was 19 when the USSR was dissolved), while Georgia (Abkhazia, Adjara), Ukraine (Crimea), Uzbekistan (Karakalpakstan), and Azerbaijan (Nakhichevan) all had one (or two, in Georgia's case). Below that was autonomous oblast and then okrub, the lowest form of autonomy. Again, Russia had most of these territories, but Azerbaijan (Nagorno-Karabakh), Georgia (South Ossetia), and Tajikistan (Gorno-Badakhshan) each had one as well.

A cursory glance at these locations would show that each one of the non-Russian autonomous republics/oblasts (with the exception of Karakalpakstan, Nakhichevan, and arguably Adjara) was engaged in some form of conflict with its parent state post-1991, and that some of the Russian republics did as well (Chechnya the most prominent). So they obviously did want to break out and not remain in the Soviet-imposed status quo.

This however then leads to the question of why weren't they successful, and why is it that there is not an internationally recognised Nagorno-Karabakh Republic, or Chechnya? I would argue that this is mainly because as these autonomous regions were just that, autonomous parts of a larger union republic, they were not given the resources needed to both properly succeed nor wage a successful war of independence. They simply were unable to break away from their nominal state, though in many cases (Abkhazia, South Ossetia, Karabakh, Chechnya for a while, Crimea if you want to downplay Russia's role) they have reached a sort of middle-ground, in that they are de facto independent/not part of their nominal state, but not able to build and maintain a de jure independent state that has international recognition.

And finally, I'll just quickly note the Siberian republics, as you mentioned two of them (Tuva and Sakha). While I'm not terribly familiar with their status, I would think it is largely due to geography and population, and a lack of interest in changing the status quo. There is simply not a lot of people in Siberia (the Sakha Republic is nearly as large as India, yet has less than one million people), and the people just don't have the legacy of struggle against the Russians (or whoever is the ruling power) to foment an independence movement.