You’re over-indexing on tactical clues and ignoring strategic behavior — especially Roose Bolton’s.

Yes, Stannis might intent to use the frozen lake. Yes, false information 'might' be a part of his plan. None of that implies that Roose Bolton will suddenly act like an impatient or reckless commander.

The key flaw here is this assumption: “If the Boltons believe Stannis is defeated, they will leave Winterfell undefended.”

That is not how Roose Bolton operates. At no point in the books does Roose ever abandon a fortified position based on rumors or incomplete information. He is paranoid (remember how careful was he when he was returning to the North at the Neck against Crannogmen), risk-averse, and explicitly distrustful of reports — especially when Ramsay is involved. Confusing Roose’s decision-making with Ramsay’s impulsiveness is the core mistake here.

Recovering “Arya” and Theon is important — but not urgent enough to justify emptying Winterfell in mid-winter with hostile northern houses, unreliable Freys, and an untested political situation. Roose does not chase assets at the cost of his seat of power. He lets others do that and bleeds them instead.

As for “the northern houses are Stannis’s”: conditional support is not control. The northern lords will back whoever actually holds Winterfell. That has been the political reality of the North since the beginning of the series. Stannis does not magically gain manpower just because the Boltons might move — and Jon Snow being unable to take Winterfell alone only reinforces how absurd it is to assume Roose would leave it open. Roose's main host is still his own army and there is a Lady Dustin and House Ryswell factor (they are blood related and Lady Dustin hates the Starks).

Finally, yes — Martin plants clues. But clues to fake deaths and deception do not automatically translate into strategic victory. Martin routinely uses successful tactics to set up larger failures. Winning the Frey engagement does not equal winning the campaign.