Earlier quoted context omitted.
It still seems illogical. Kharkiv had a huge Russian population. I had some friends from there who sighed "finally!" when they heard Putin started the invasion. They truly believed he would end the years-long conflict and finally they will live in peace on a territory more or less Russian. Today, there is no single Russian in Kharkiv who would support Putin or even believe what he says. He betrayed his own people.
The entire invasion is illogical. There was arguably some sense to Crimea - there is no sense in this invasion.
- Crimea wants to control the canal from the Djnepr, which had been shut off by Ukraine.
- Russia wants the coast line to control more gas fields in the Black Sea. The U.S. of course also want to control them by proxy (see Burisma/Biden). Snake island also controls some of the fields.
- Rightly or wrongly, Russia is scared of NATO expansion.
I think it may be the case that the great Russian empire theory is a pretext that is put forward by Putin for his own population and eagerly taken up by the West for propaganda. Attacks on Kyiv (stalled) and Kharkiv could be diversions from the south. As noted above, the Kharkiv attack does not make sense though.