You would have to assume that the Institute would even WANT to take over. I get the strong impression that the Institute changes direction every time they change director. Each new director drives the Institute toward what ever personal goal he/she has in mind. The rest of them just plod along doing "science for sciences sake", and dreaming about what they would do if THE were in charge.
Read enough of those old tapes, and you see several different viewpoints at work in the Institute over the years....some hostile toward the surface, some ambivalent, some completely apathetic. The "Speech" you give, that was crafted by Father, pretty much says that the (current) Institute has no interest in the surface.
Add to this, the fact that Synths are still pretty much in a Beta phase, and the Institute hasn't really made all that many of them, other than to replace the ones that keep going missing, or are destroyed while on surface "recon"...