Rudimental solos are for snare, usually. It think it's great to learn them, and keep time with the feet. Why there aren't etude materials like this for drum set is actually a deep question.
Any book of drum set solos (though I admit I haven't seen the Ted Reed book mentioned earlier) is bound to be dated, or devoid of musical context, or both. What books I have seen that attempt to apply rudiments to the drum set are pretty lame, only scratch the surface, and provide few practical combinations, IMO.
And although every rudiment can be voiced on the drum set a number of ways, I don't think it's wise to go through every possible permutation, without regard for which combinations lend a musical sound, or for whose executions are logistically fraught with issues. Why not learn a few combinations and voicings that "work", and learn from there? To find them, though, you'll have to check out drummers who have done that sort of thing. Steve Gadd has great "licks" derived from paradiddles, ratamacues, six stroke rolls, paradiddle-diddles, and other phrases. Steve Smith also has some signature licks with double paradiddles, paradiddles, and what Bill Bachman calls "puh-duh-duhs".
Todd Sucherman has a section on his DVD called "tribute fills" where he explains some pretty crazy stuff, and from whom he heard/learned it.
The book Studio and Big Band Drumming by Steve Houghton develops the idea that the right hand "leads" around the drum set, and the left hand "fills in" the remaining spaces (very handy stuff if you have to quickly nail a horn figure!). Yes, you could surely describe those combinations in terms of ratamacues, double paradiddles, etc., but it would be unnecessarily confusing, and prevent you from improvising in such a way that goes even beyond rudimental thinking. Peter Erskine talks about developing a series of "signposts" for his fills, and then adds notes and phrases around them, without particular attention to what rudimental combination he's using. Other drummers might be more deliberate, of course.
I guess that, as a teacher, I wouldn't want you to play a certain rudimental "routine" on the kit. Rather, I would prefer that you explore some combinations and voicings, and deepen your "drum set vocabulary". Timekeeping and transitioning between the drums will happen as you become familiar with the new combinations. Give yourself some time to just play through it, then throw on the metronome and play slowly for a while, and then at a variety of tempi. And yes, write out the combinations you are learning, and especially the ones you like!