My impression is that putting a specific number puts more emphasis, because it’s like we’re not saying it lightly (as if we actually weighted the things we have to do–metaphorically speaking of course) whereas “des tonnes” is more vague and perhaps less convincing.

I would guess French people love the ‘R‘ sound, that is so difficult to pronounce for strangers, because we can accentuate it as much as we want.

‘TR‘ is even more aggressive, it shows how what comes next is important. Add this to “Trois tonnes” ; you gotta admit that the double T sounds great, doesn’t it ? Maybe it reminds us of ‘T’N’T‘ !

Another example would be :

J’ai trente six mille choses à faire !

We like to use this as well.

Ah I think there nothing to see here, it just means ‘a lot’.
Dix tonnes, trente tonnes, dix milliards de trucs, vingt mille trucs, trop de trucs, trop pas de trucs (ironic).
All the same to me. Emphasis is for a humoristic tone (tone, tonne, what a pun!)