I would think historical czar-dom does in fact play a significant role in explaining the state of Russia today. As it is, Putin very much appears to aspire to czar-dom, and it's not impossible to believe the Russian people will grant it to him.
I don't support blaming current woes on the past however, as it carries the implication there's nothing can be done to improve things now, which is almost never really the case.
"Putin is trying to be a czar, this is bad and Putin is bad"
Vs
"Putin is good, everything wrong with Russia is the fault of those perfidious czars! Citizens are reminded that any criticism of Putin is illegal czarist sentiment, and to report all czarist sentiment to Putin's secret police. They know where to find you."
When South Africans realized their public funds were being looted by the Gupta family, and the latter paid PR firm Bell Pottinger to come up with a campaign to blame everything on white people ... that starts looking a little more like the latter situation.
But you could argue that the ANC only gets away with blaming everything on white people because the narrative that the country was historically screwed up when it was run by them actually has some credibility to it.
Again, not defending that blame-culture at all, it's completely counter-productive.
I don't "blame" either! But I would expect the period of Mongol rule left its mark too, particularly in how it led to the rise of the czardom, though I'm way out of my depth there.
Putin is a nationalist dictator whose foreign policy seems to be driven by some combination of a persecution complex, an inferiority complex, and a Mafia-like need for "respect", and perhaps to distract from the domestic situation. I do not think there are any signs that resurrecting the Czarate is one of his motives; it is merely that they have resulted in something that resembles it (or the Soviet Union without the nominal socialist theory, for that matter.) One might as well compare him to a Byzantine emperor or a shogun.
