Russia doesn't want to take areas it doesn't have popular support in. It just wants to do exactly what it did to Georgia, punish its opponents and annex the friendly bits.
Everyone's doom-mongering, but Russia has a clear strategy.
Russia's strategy thus far has been to walk around and ignore the east, and instead attack big cities like Kyiv or Kharkhiv.
Russia's biggest gains have been their amphibious assaults from the south / Crimea.
But all attacks from the north and east have been lackluster this far. And bypassing the fortified line means that Ukraine has the ability to attack those overextended units in the rear.
Russia seems like they are rushing this. I think a slower, more deliberate advance would help the Russians here, they really shouldn't be trying to take cities yet just 3 or 4 days into the conflict. The Russians should have tried for air superiority and strengthened supply lines for a more drawn out assault.
Of course, I don't know the Russian supply situation. It could be that these units know they don't have enough supplied for a drawn out battle and are rushing because they need a quick victory?
The other theory is that Russia is doing the Cannon fodder strategy, purposefully sacrificing weaker troops to the enemy to eat up Ukraine's ammunition (javalin missiles or whatnot).
Putin is invading a country that nearly unanimously opposes him with troops that thought that they were just there for an exercise and don't really want to invade their brother country.
Actually, the blue areas turned out a majority vote for https://en.wikipedia.org/wiki/Opposition_Platform_%E2%80%94_...
There are definitely some pro Russian electoral districts.
Here is the map that shows the support for the pro Russian party.
https://upload.wikimedia.org/wikipedia/commons/5/56/%D0%95%D...
There are very few districts, that have more than 50% support for the pro-Russian party. You also can't assume that the voters that voted for a pro Russian party want to be part of Russia and support the invasion.