What are the structural and chemical characteristics that make tetracycline uniquely broad spectrum? I understand it acts on the A-site of the prokaryotic ribosome, but there exist many ribosome-targeted drugs that are relatively narrow spectrum. I assume it has something to do with its permeability in several different classes of bacterial cell walls and/or membranes - but I can't find any literature that links some characteristic part of its structure to this permeability.
This is a tricky question. First of all, I wouldn't call tetracyclines "uniquely" broad spectrum. They are broad spectrum, but there are other drugs with equally broad coverage absent acquired resistance (e.g., imipenem and chloramphenicol), and current clinical use of tetracyclines is limited somewhat by the spread of resistance. They don't have any major class limitations, though. Again, absent acquired resistance, tetracyclines are effective against gram positive aerobes and anaerobes, and gram negative organisms, as you seem to be aware.
Tetracylines access their target on the 30S subunit of the bacterial ribosome (blocking the A site, as you said) by passive diffusion through porins of the outer membrane of gram negative bacteria, and active transport across the cytoplasmic membrane of gram negative and gram positive bacteria, but the mechanism for that active transport is still unknown. This is quite a good review despite being quite old. Now, some 30 years later, we have still not identified the exact mechanism, as you can read in Goodman & Gillman's most recent edition (13th), Ch. 59, crossing the cytoplasmic membrane requires metabolic energy, but the process is not well understood. Because of this, we can't point to a specific structural or chemical characteristic. For chloramphenicol, it would be the ability to diffuse across the cytoplasmic membrane; for tetracycline, we don't know. I suspect part of the reason for this is that people are now more interested in pumps removing tetracyclines from the cell than those that accumulate it, so it's more of a question about research priorities than some transport mechanism that is particularly difficult to detect.