For UIUC, you can apply without TSE score. If admitted, they will ask you to take SPEAK test which is very similar to TSE but held by the university. However, you need to bear the risk to be kicked out of the program if you can't pass SPEAK in the first year.
I did ask the same question when I applied for schools. Most schools kept silent to this question so I guess it's not strictly required for the admission purpose but you will need it to be a TA after admitted.
I guess you can still apply without TSE score, but I can't guarantee you won't have any problem in the process. One thing is for sure: don't argue with secretaries. It's usually a waste of time.