Then they'd probably stop happening, because local authorities are mostly strapped for cash.
There's no such thing as the road budget. There's the trunk roads budget and there are local authority budgets. But there are virtually no level crossings left on trunk roads.
Then the DfT should pay for them out of it's trunk road budget regardless.
I don't see why the railway should be paying to resolve a problem primarily caused by motorists, and what's more I never will.