I have never seen the exact timelines for things laid out.
I don't think any public timeline is available, but statements from all sides make clear (i) Alpine wanted Alonso not Piastri, (ii) Alpine had no contractual claim on Piastri when he chose to leave, and (iii) Piastri had a McLaren contract either signed on in final negotiation prior to Alpine announcing him
Even in the latest Drive to Survive they appeared to make out that Oscar did Alpine the dirty, without clarifying that he in fact signed for McLaren before Alonso left.
They mostly quoted Szafnauer who said a whole bunch of frankly outrageous things both at the time and to camera on DtS. The person whose reputation should have suffered most from this is Szafnauer.
More generally, even if he had free choice, I don't think it was a bad choice for Piastri. McLaren had probably the better car last year, a new wind tunnel coming on line soon, and a lot of promise. Even this year I think they're better than people realise, I suspect in a clean race they're about level with Alpine.
I think Alonso jumped ship because Alpine were trying to secure Piastri
Alpine were planning on parking Piastri at Williams until Alonso aged out of the sport, although AFAIK nothing was signed on that. They offered Alonso a one year contract with an option to extend, but he didn't want it. It's pretty clear Alpine were prioritising Alonso over Piastri and thoroughly deserved to lose him after all the crud they pulled. Szafnauer's histrionics after the fact were even worse.