I think you are on the right track with your idea of continuing music without pauses. Many people have word sounds for different rhythms. For example for triplets: "one and uh two and uh". Having other words can be even more fun.
Here is a quote from a website that mentions Michiko Yurko's book Music Mind Games:
"I’ve been teaching how to count rhythm to most of my students using Michiko Yurko’s genius method of naming note values with easy and fun to say words. I highly recommend her book Music Mind Games for all music teachers and home-schoolers and interested parents..
For example, a one beat (quarter note) is called BLUE.
Two eighth notes are called JELLO.
An eighth note triplet, where the three notes are played in one beat is PINEAPPLE.
And four sixteenth notes is HUCKLEBERRY.
This is so much more fun and easier to remember than when I was in school learning, “one -eee- and – ah.”"
That quote is from this website:
How To Teach Rhythm to Beginning Music Students | Park Slope Music Lessons
Clark Terry wrote a book "Introduction of Sounds to Rhythmic Patterns" (out of print) that has wonderful vocalizations notated for many jazz phrases. One can hear those sounds when Clark would scat sing. The are notations of what various jazz phrases should/could sound like.