Relatively few Russians, in my estimaton, would recognise the concept of 'draining the swamp.' The existing system is just one variant on how they're used to living.
Simon Jenkins has it right in the above-linked article where he says
'When I asked a Russia expert what he thought would be the true tally of electoral support for Putin’s dictatorship, his view squared with this survey. He suggested it would be about 60%, though lower in Moscow and St Petersburg. This sounded much like my visits to Moscow in the post-communist 1990s. Russians would concede the virtues of western democracy, but they pleaded the more urgent need for order, security and prosperity.'
That chimes with my experience in the late 1980s and early 1990s. Almost everybody a westerner pissing about in Moscow was likely to get to know was the type to recognise that the USSR couldn't go on., but it soon became clear that most didn't really believe that their future could lie with the west, nor that anything in their country could fundamentally change. I soon came to recognise that most didn't really want it to as it was all they knew. And even the most 'liberal' of them were wary of the prospect of societal chaos.