They've done that to some extent via Borghetti, didn't exactly work out well with their main competitor responding quickly.
As said previously, Star Alliance would've been a drain on VA's budget with the membership fees, and possibly further draining to other airlines via FFs charging VA for lounge access at member's lounges. VA's VFF changes didn't impact as much as some on this forum as thought (revenue did slightly increase for VFF), so it'll be interesting to see what the FY26 results for VFF end up as is.
In addition, Star Alliance does require a "full service" aspect on their services, and VA is a hybrid budget carrier which wouldn't really qualify with Economy being LCC+ with their base fare being "carry-on" only with complimentary water and tea provided.
I don't see the larger shareholder Bain spending the Capex to re-apply full service on some of VA flights after rolling VA back from full service to a hybrid LCC.
Qatar Airways has their own (political related) issues in the Middle East to go through before doing anything furher with their 23% stake in VA, and there's still higher chances of VA being dragged into a customised QR alliance rather than spend the CapEx on general Alliance membership.