This is not a democratic court; it is a royal judgment.
As Jon's nominated proxy, Sansa carries all the powers of a "King [queen] in the North". It is her sentence, and she does not need to seek the approval of the lords to carry it out.
Littlefinger's behaviour does little to redeem him for his actions, and in some cases he outright admits the crimes levelled against him (such as the murder of Lysa Arryn, although he professes a benevolent motive).
In the example above, as with all his accused crimes, the person he has to convince in order to be pardoned is Sansa Stark; and as she is accusing him of his crimes, this isn't going to happen. Sansa has her own direct experiences of Littlefinger, along with the counsel of her sister and the three-eyed Raven... and Baelish co-ordinated events that led to the rape of the proxy Queen in the north.
All Sansa needed/wanted was a recent crime to provide the context for a fresh trial. Littlefinger's conspiracies were easily tantamount to this. The lords would have accepted her judgement even if Baelish hadn't confessed to his crimes, tried to flee justice etc.
Edit: for the question of why the Knights of the Vale belayed the order of protection; Little-finger just admitted to murdering Lisa Arryn, their Lady Regent. He also professed his reason to be "to protect you [Sansa]", a claim which Sansa refutes and counters that he was acting in self interest. Regardless of motive, he just admitted Regicide. They were never his own soldiers, they were house Arryn's; and he just confessed to a Capital crime against that house.
To be clear: No one in that room is going to come to Little-fingers defense unless they are compelled (through honor, duty or obligation) to react. Baelish was a necessary evil these people were being forced to endure. Little-finger believed himself to be hated, but indispensable. His power of statecraft was, however, always based on manipulation/subterfuge.
The Starks simply asked him to explain himself openly to a room full of accusers; denying him his usual method of individual manipulation. It for this reason he asks to speak to Sansa 'privately', and it is for this reason he is plainly rebuffed. Without being able to spin lies and turn actors against each other, he is powerless to defend himself against his crimes; and even admits some of them.