In my opinion Eltsin’s Russia in fact differed not too much from Putin’s Russia. Eltsin’s Russia just did not have the resources to implement imperial look like policies that Putin is able to implement, though Eltsin tried. Eltsin just before the USSR break up threatened to review the borders between the USSR republics, should they secede from USSR. Eltsin launched war in Chechnya in December 1994 (even though real operations started long before that).
So you are right, the West was much more benevolent towards Russia when it was weak, because then it looked much more hopeful and much easier to manage. There certainly were mistakes that the US and the West in general made in Russia. One of them was that they were not hard enough on war in Chechnya, Clinton saying something like: if we had the same problem in one of the US states, we would not let it secede.
NATO along the Russian borders is in my opinion way of ensuring that Russia does not reincarnate USSR, which is good in my opinion, because if Russia manages doing so, the creature will be just as nasty as it used to be long ago. The purpose is to incorporate Ukraine and Georgia as key (and most feasible) countries, because if the west doesn’t incorporate them into something, they might potentially get under Russian sway. That is the simple logic behind these moves. But also both Ukraine and Georgia have this wish to join NATO for one reason or another and it is difficult to deny them this right just because Russia doesn’t like it. As the West cannot offer them EU membership, NATO is apparently meant to be sort of relatively cheap substitute of it.
Russia presents real threat to the US and the West in general and to a few other regions and countries. I have no least doubt that if it weren’t watched closely Russia would have invaded Georgia long ago for example. Russia clearly plays Iran against the West and it tried to play China against the West (which did not work, of course). The West had been talking about danger of a “dirty bomb” being used by some middle eastern terrorist group until they got it from Russian security services delivered right to London (Litvinenko’s case). If a country has an oppressive regime it inevitably spills over its borders. Remember Iraq’s invasion of Kuwait. So the West had better be alert, rather than rely on the assumption that “Russia is not a threat”.
As soon as Russians regained control of Chechnya, suppressed freedoms in the country, murdered a few journalists and businessmen or put them in prison it immediately became vehemently hostile to the West (and not only) and assertive in its foreign policy. It is difficult to deal with Russia and if you notice it doesn’t practically have friends, even Lukashenko is nowadays releasing political prisoners in order to establish better relations with the west. At the latest conference I visited in Georgia it was the Armenians, the assumed Russian allies, that were most critical of Russia.