[2023] PlayStation Store Spring Sale not get the Pack

Yes, you have to login to War Thunder from the console to activate it. It can be any console (PS4 or PS5), as long as its owner will let you login to your PSN account from it (and has War Thunder downloaded).

It's working as intended. Premium time included into vehicles packs in PS Store is only available when you play on the console.

Only premium account purchased in-game for Golden Eagles is synced between console and PC.