I just tried the link you posted. It doesn't appear to follow roads, it just draws straight lines between clicked points. ridewithgps.com has way more route drawing advantages.

I think it would be tedious to make routes on a phone. Picking the correct roads really needs a mouse pointer for accuracy and speed. And it's typical that the route needs editing, either as you make it, or after it's done the first time. All this is much easier with a mouse.

I don't know anybody that uses mapmyride. Years ago, it started as a route planner, but went into a social media direction. The routing isn't it's focus.

Yes, ridewithgps.com is by far the most commonly used route planner for cycling. It's free, just create a login.

To make a route in ridewithgps:
Browse to the starting location on the map. Click the starting point.
Now click farther down a road. It follows the road, and draws a red route line. The elevation chart at the bottom updates as you go.

There's Undo and Redo if you want to change to a different road.
Or just drag the red line over to the new road.

You can drag to select part of the elevation graph, and you get stats on that section of the route -- total elevation, distance, avg and max grade.

When done, click Save, and give the route a name, to store it in your Routes history.
Then you can view the route and Export a "tcx" or "gpx" route file into your GPS bike computer or phone.