Semplice - I don't know much about the takedimi system but there are many who love it - it seems fairly widespread in the USA. I think that the 'ta' is always used for whatever falls on the beat - ta-di is the quaver subdivision and ta-ke-di-mi the semiquaver subdivision.
Sorry, but I spent 11 years of learning the piano with teachers who thought that the way to instil pulse in me was to say 'One and two and...' loudly. I'm afraid this approach was unsuccessful.
I find rhythm names just so good. I hate 'tea', 'coffee', 'caterpillar' etc. with a passion! But it's the relating to the pulse which is the key to successful rhythm reading and understanding.
I rarely have children clap rhythms. Clapping can be very, very unmusically-done (I love the Dalcroze way ) - the rhythm names just totally keep focus and clarity and accuracy - particularly if the child performs the pulse in some way whilst speaking the rhythm names.
Knowing the theory doesn't help in the slightest with the actual performance of the rhythm. So two quavers equal one crotchet? Er…??? I could have told you, as a child, how many demisemiquavers were in a minim, but wouldn't have had the first clue as to how to accurately perform either of those rhythmic elements (OK, I was obviously a very slow learner . But there are many others out there like me...).