Based on my time in China in both professional and casual settings, I would interpret the term 着急 here as meaning "rushed." "Hurried" might also work.
My translation of your sentence would be
It can be said that we've already begun to push the transition from exam-oriented education to quality-oriented education, but this transition is very difficult. It is a painful and slow/very long/endless/arduous process. We should not/cannot be too rushed/hurried.
To illustrate, I give an example:
My wushu teacher might tell me
which translates to
This move is pretty difficult, don't rush it, do it slowly, otherwise you'll easily get injured.
As you've pointed out, very often people and dictionaries will construe 着急 to mean "worry" or "anxious," but I feel like using "rushed" or "hurried" is more in the spirit of the sentence, given the context of a "painful and arduous" process.
有道词典 contains some examples where "hurry" and its derivatives are used. In the entry for 着急 under more examples
Israel seems in no hurry to prevent that; nor does Mr Abbas, nor their
local allies, the leaders of Egypt and Jordan.
Investors like to wait ("flip another card over") while you want to
"No," the waiter who was in a hurry said, rising from pulling down the