Oh, I really like this! The whole part from Minerva's musings about her possible own role in the making of Severus Snape until her realisation that Severus wants punishment (which you build up beautifully!) is everything I love to see in Snape fic -- complexity, moral dilemmas, and ultimately the impossibility to judge in terms of essential good or bad. Your Severus is both a human being with agency and a drive of his own and a victim of his circumstances.
There are any number of passages that illustrate what I mean, but these are the ones I found among the strongest:
Yet over the years she'd discovered that the situation between Severus and the Gryffindors had been more complicated than she'd ever realised at the time, and she had come to question her own role. Had she allowed herself to be swayed by Severus's unprepossessing exterior or by his adolescent sullen posturing? Whatever the reason, she very much feared that she had not done right by Severus the boy. Minerva straightened her shoulders and took herself sternly in hand. Such regrets were useless. Not even a time-turner would let her unmake Severus's difficult past. -- Perfectly Minerva, insightful and sober.
Then there's the scene with the three of them, with Albus's criticism of Severus's teaching (I'm glad someone at Hogwarts actually paid attention to that; from the books it seems as if he'd been given a free rein), and of course Minerva would acquiesce to his proposed remedy, while Severus would get a bit worked up.
"You most certainly need something!" she shouted. "Look at you, behaving like a foul-mouthed, emotional teenager. Do you think you have a corner on unhappiness?" -- Oh, brilliant! Of course he has reason to be angry, but yay for Minerva giving him some perspective.
"Yes! No! No, of course not. But I wanted...I mean, I did want..." His eyes glittered, and as he let go of her to wipe an angry hand across his face, she knew, all at once, what he needed. [...] He wanted that clarity, that firm line drawn between wrongdoing and atonement, a line one could step over, leaving the mess of error behind, to stride into the clean light of a new start. -- This. So much this!
The way you wrote Minerva's POV during the spanking is also intriguing. You see her Gryffindor ambition, and also a motherly/fatherly element to it, but in a way that doesn't clash with the arousal that she feels and denies (at least so long as she's in Severus's presence.) And it is simply great that Minerva is aware that this was but one step in a long process of atonement, much of which will not involve her because while she can help, it is not in her power to fully redeem him.
Of course she would do what she could to support him, though she feared it would not be much. Whatever forgiveness he found could not come from her, not from someone he'd never wronged and who had sufficient sins of her own.