> Why should the Ukraine be so important?
Side note: you've used the old-fashioned (and Putin-favoured) english nomenclature for Ukraine as "the Ukraine" in the first sentence, and the Ukraine-preferred nomenclature in the second.
This is actually the crux of the issue in two sentences. Ukraine is a sovereign state, but Putin politically asserts it is merely a territory they control, harking back to a pre-WWII time; Ukraine was a territory regularly divided up and under the control of different neighbouring states as a bargaining chip, gift, settlement, or conquest.
This is I guess why Brzezinski talked about it as it was and as it is. Ukraine used to be passed around and fought over as a set of territories without a home. Post WWII, it is a nation state and its existence as a nation state historically perturbs its neighbours, most notably Russia.
Ukraine still has all the geographical significance it had. It's the second-largest country by land mass in Europe, it is fertile (a nearby breadbasket, geopolitically), it is also mineral rich, etc.
So Brzezinski wasn't wrong to say it was still a source of conflict, a prize, but now being an independent nation state it has the right to defend itself and enter alliances. That makes it a risk strategic point -- even perhaps a pivot -- in a future conflict.
> Why could Russia not be an empire without Ukraine?
If Russia is to (re)build an "empire", it needs Ukraine to do it, for all the reasons above.
Bold to say Brzezinski wrote nonsense, though. Good for you: put yourself out there as an international statesman.