That's another good question, which again deals with unambiguous signaling. Remember that molecules have not only size in space, but also a shape and a pattern of electron density. So, yes, you could make a protein as big or bigger than T3, but will it fit into the necessary active site the same way? And will it electronically interact with the active site the same way?

For peptides, the bond angles of the main chain are given by something called a Ramachandran Plot, which works out the most likely shapes a peptide bond can form in space. In this case, maybe you could have a protein with a pair of tyrosines that could meet up and maybe look kind of like T3? Well, the necessary bond angles make that unlikely, so such a shape would not persist in space for enough time to matter. Even if it did, you'd have the rest of the protein hanging out at weird angles, preventing a good fit with the intended receptor.

Finally, even if all that worked, there's still some intentional interaction with the iodine substituents necessary to induce a tight fit. You could get a couple tyrosines in there to fake the rough shape, but how are you going to get protein parts in there to fake the iodine interactions? There's no way. (A) nothing else is even kind of shaped like those iodine substituents, and (B) even if there were, you couldn't stick them into that binding site without popping the whole protein out. Thus, the iodine groups help ensure that only T3 will reliably fit into the intended receptor site.