A phrase that seems redundant

A phrase that seems redundant

The passage below is the last passage of The Library: A Fragile History by Andrew Pettegree.

Although there were few who could match Niccoli’s generous lending, it was expected that humanist scholars would open up their libraries to others. This was one of the crucial tenets of book collecting that the scholars passed on to their patrons, the political and ecclesiastical elite. The humanists could point to many examples from classical Rome that indicated the lengths to which Roman generals and emperors had gone to amass great public collections. That these libraries were circumscribed by a limited notion of openness we have already seen, but to the Italian princelings, who believed their world to be Rome incarnate, the building of impressive libraries was a challenge that they readily accepted.

This paragraph has four sentences.

First sentence says humanist scholars’ opening up their libraries to others.

Second sentence says humanists passed this openness on to their patrons, the political and ecclesiastical elite.

Third sentence humanists explains (probably to their patrons) the lengths to which Roman elites had gone to build great public libraries.

Fourth sentence has two complete sentences between ‘but’.

The first half says libraries Roman elites build were not open to all but the select few.

The second half says BUT Italian elites who believed they were successors of Rome were willing to build great libraries.

Here I think the first half of the last sentence seems irrelevant in this context. It is because of ‘but’. This ‘but’ is used because of the third sentence.

In the third sentence humanist scholars tell Italian elites how much efforts Roman elites had had to make to build their libraries.

If we omit the first half of the last sentence, the third sentence logically agrees well with the part after ‘but’.

It takes great efforts to build great libraries like those of Roman elites. But the Italian princelings were willing to take those efforts.

Do you agree with me? If not, what’s the use of the first half of the last sentence?

Thanks in advance.