Desperation or practicality? The whole point of a TD is that it's a mobile AT gun. They used to tow AT guns with half-tracks. Then they figured why not just put it on the vehicle itself. On the vehicle the AT gun is far more mobile and can retreat quickly, saving the gun crews the time to set up and pull down the gun between engagements. Most tank destroyers, Allied and German, were rather lightly armored. Some TDs in German service were exceptions like the later StuG and Jagdpanzer III/IV series (and the very rare Jagdpanther and Jagdtiger), but there were other fully tracked German TDs that were lightly armored, open topped affairs.
Panzerjäger I
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Panzerjäger Marder series I-III
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Nashorn
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Lightly armored German TD's were either stop gap measures, or specifically built to engage at standoff range on the Russian steeps (Nashorn).
It is entirely fallacious to assume that because they built light TD's cobbled together from various tanks and guns, that that is exactly what they wanted.
The T-34, though killable by a Panzer III, gave the Germans fits early in the war. They were desperately trying to get more firepower to the front, even going so far as to actively try to capture Russian 76.2mm AT guns, since they could be rechambered to accept German PzGr. 39/40 ammunition, or left chambered in 76.2mm and still provide acceptable performance.
While the Sd.Kfz. 251/22 met with some success when properly employed, in the same manner as a towable AT gun, it should be noted that this was really the only way they found any success. Point being it's not really fair to include them with tanks.
They were built to function in the same way as towable AT guns, as you yourself noted. And mostly that's how they were used.