My understanding is that the word "rider" has entered into common usage in Japan with a slightly different meaning - more like how we would use "biker". Their understanding of it has certain implications of behaviour and attitude rather than just literally being "someone riding on a bike". Lead characters like Shinji, Ryotaro or Gentaro certainly don't fit the biker stereotype, so I'd argue that the series has long since violated the spirit of the title, even if they all physically sat on a saddle at one point.
There are quotes that Kuuga's producer said he wanted to re-write how people saw Kamen Rider. Agito brought in many tropes previously more typical of Metal Hero and other toku franchises. More recently, interviews with Urubuchi have made it clear that the current situation is that Bandai comes up with a toy line and the showrunners basically just make a show to promote it. We haven't had cyborgs, most Riders are in no way modified to become Riders and some use biological transformations so you're looking at their actual changed faces rather than a mask. The tone has changed from dead serious to full camp and back around again (sometimes in consequtive episodes). We've dropped bug eyes sometimes. We've had shows that haven't used the words Kamen Rider outside of the title card. We've had villains with goals completely different to taking over the world. We've had villainous Riders. Heck, we've had a Rider that was completely CGI.
Just about every aspect of Ishinomori's formula has been rendered optional over the years. So should the bike be any different? Not for me personally. Other toku shows have had bikes. Various Sentai, Metal Heroes, other Ishinomori shows like Kikaider and Akumaizer 3, and so on. I care about watching enjoyable toku shows, not what the name should stand for. I'm not saying you're wrong if you do care, just that it's not so important to me.