A Princeps' rank does seem to be relevant to the type of Titan they command. A higher-ranking one, such as Princeps Majoris Zarha Mancion of the Legio Invigilata, controls the largest type of Titan, the Imperator-class Titan "Stormherald". Mancion commanded a full third third of Invigilata during the Third War for Armageddon, and her rank of Princeps Majoris is in the top three most powerful ranks of Princeps in the Collegio Titanica. Most likely the reason for such a high-ranking member of the Legio being in command of the strongest and most powerful engines is, as you suggested, because they provide the most protection.
However, it does not appear from anything I've read that an individual Princeps actually gets promoted to a higher rank or that they ever leave the Titan they're bonded to before their death. A Princeps and its Titan are one and the same, more or less, due to the sheer amount of mental (and often physical) bonding that is required to control such a massive war engine and become symbiotic with the Titan's own sentient computer system. Removing a Princeps would be difficult at best and deadly at worst, and as it takes years to bond with a Titan, it wouldn't make sense that the Imperium would simply transplant different people to heavier grades every so often even with the much-expanded lifespans of Mechanicus.
That said, at some point or another, it would certainly become necessary to replace a Titan's Princeps. I would think that the Collegio Titanica would draw from its newest members, assuming they were found compatible, to command one of the Titans in need of a human component. It may be that when a new Princeps is required, the ones with the highest scores on whatever testing methods the Collegio make the candidates undergo are automatically bonded to the best and strongest Titans as they are made available rather than giving an exceptional candidate a weaker engine or risking transferring a Princeps from Titan to Titan without a guarantee of success. This seems to be the most likely explanation. The WH40K wiki states that one in ten million candidates (or fewer) may be qualified to be considered as a Princeps and even fewer succeed in completing their training. Since their numbers are so few and far between, easily some of the rarest individuals in the Imperium, it would only make sense that a Princeps is bonded to single Titan their entire life.