https://www.mapquest.com/ uses Mapbox in the background. https://maps.mapcat.com/ uses OpenStreetMap vector tiles. On mobile I use https://maps.me/download/ (free, offline use possible).

MapQuest feels a bit strange with the huge amount of advertising. I'd much rather pay.

OsmAnd can do this. It's generally recommended as the OpenStreetMap app, but almost everyone uses it with offline maps. If you click past that and enable the built-in online tiles addon, you can have tiles. Search and navigation, I don't remember seeing it in recent versions but there are so many options, I'm almost certain it must be in there somewhere. Search for custom or online navigation provider in OsmAnd and you should be able to find it.

As you see, it's not exactly built for this type of usage, but it is supported and the app is generally considered as the best alternative out there, so it might be worth a try!

For good search and routing, you need a high-quality map data designed to deliver professional find, routing and navigation services. I recommend you take a look at developer.tomtommaps.com