Giving cash is not tacky. Wanting to get cash is not tacky. Sending your wedding guests an invoice with the invitation is what’s tacky.
BTW the gift registry originated so that guests could buy things from the department store like china or flatware place settings that would be the same pattern that other guests might be buying, so everything would match. You would register your pattern. Now it’s just a “gimme” list, what with all manner of stores establishing gift registries.