route planner on openstreemap.org: Use on own websites map?

Can I include the route planner (shown on www.openstreetmap.org) in my own map? I am using leafletjs but its plugins don’t satisfy my needs.

If not, is it possible to offer a link to www.openstreetmap.org which opens the site with its route planners field pre-filled? Maybe through REST API?

Why don’t you try to include a solution directly from https://www.graphhopper.com/ or http://project-osrm.org/ or others listed on https://wiki.openstreetmap.org/wiki/Routing → ONLINE-ROUTING in your web site?