Eliza, bulb alliums are persistent but do expire after 3-4 times removing leaves and some stem with a trowel, although not removing the bulb which as you say is so buried.
The bindweed needs to see your trowel too, every 10 days in summer, at least. Little and often.It’s not a losing battle but you never “win” either.
I have the same here with a wooden fence between one of my borders and a neighbour’s bed full of ivy and some bindweed, it is constant “edging” to keep them at bay.
