You might want to look into https://cycle.travel/ (my site!), https://brouter.de/brouter-web, Komoot or RideWithGPS - all do OpenStreetMap-powered bike directions.

Getting bike routing right is really challenging, partly because often there genuinely isn’t a good route, and partly because standards and infrastructure (and OSM mapping practice!) differ from country to country. I love the challenge and the feedback from users when it works well, but it’s a massive task.