On paintings, portraits and plots

I've written before about the role played by paintings in the magical world of Harry Potter, particular the portraits of former Hogwarts headmasters that hang in Dumbledore's office (sorry, I can't bring myself to refer to it as Snape's office).

It is clearly established that people captured in paintings can communicate with the living people who  wander past them. This happens all the time. It is also clearly established that the characters in the paintings actually carry on linear lives, lives that are situated within the same chronological context as the living beings.

These painting people experience events at a particular moment in time, then move on to other events while carrying memories of the earlier event. The Fat Lady recounts several times, for example, how she was attacked in book two, and finally identifies her attacker as Sirius Black. In the final battle for Hogwarts, the painting people carry information from place to place in the castle.

And the portraits in the Headmaster's office, in particular, appear to carry the memories, the attitudes, the opinions of the living people they represent. They are an extension of the earlier headmasters and appear to add the experiences and memories they accumulate as painting people to the experiences and memories they held when they were living people.

Does that all make sense?

Okay. So why would Harry ever feel abandoned by Dumbledore in Book Seven? Why would his first goal not be to break into the Headmaster's Office and have a long talk with Dumbledore as a painting person, to obtain the answers to all the questions he has?

If the painting Dumbledore is an extension of the real, formerly living Dumbledore, why would Harry, Hermione and Ron not try to find a way to get to the portrait and find out how to destroy Horcruxes, where the remaining Horcruxes might be, what the deal is with the Hallows, etc.?

And why wouldn't Snape have found some way to help Harry and the gang gain access to Dumbledore's portrait without raising suspicion?

Again, I am concerned that, in creating these magical portraits in so much fascinating detail, J.K. Rowling has set a trap for herself that actually ends up undermining the effectiveness of these wonderful stories she's written. Wouldn't it have been possible to tell the stories without making these portraits so fully functional as to undermine the plots?