When it comes to cycle paths you need something like on Park Lane in Liverpool, where it's built into the road, but separated from it by its own kerb. It's the only way to stop motorists from slipping into the cycle lane.
That said, I've been walking along pavements more than once and found a cyclist coming straight at me, when there is a perfectly good cycle lane clearly marked on the pavement, mere inches away
Cycle along Oxford Rd in Manc quite often, which has cycle lanes like that. Nearly every time, there's a private hire driver sat across it, right where the entrance to a kerb bit is. Once even saw a car parked (no one in it) straddling the double kerb stone.
The cycling infrastructure doesn't work as long as there's twats driving around. And if there wasn't, there was no need for specific cycling infrastructure.