It does mean that they need to find somebody who's universally tolerable among Democrats, which is a very tall order. In this case, they're not objecting to her background, but to her policies against large banks. They'll object to anybody with those policies, regardless of their baggage. So it comes down to a matter of negotiating who can be made acceptable to a party who has to act unanimously.
That is an immensely complicated intra-party matter of persuasion, negotiation, horse-trading, and occasionally bullying. Sometimes the administration will win; sometimes they'll lose. But they can't afford to write off everybody that Republicans think has "baggage", because Republicans will never vote for them and so there's nothing to negotiate with. They'll start instead with people the administration thinks are qualified, and do the hard work from there.