Two completely different mechanisms, and that is why more time did nothing.
Meat becomes tender because proteins change — they denature and connective tissue converts to gelatin. That starts happening at quite low temperatures, so a steak at 55 degrees is genuinely cooking, and a tough cut held for many hours at a low temperature becomes tender.
Vegetables become tender because pectin breaks down. Pectin is what glues plant cells together, and it barely degrades below about 83 degrees. Below that threshold you can hold a carrot for a day and it will stay crunchy, because the reaction you need is essentially not running.
That is the whole answer to your experiment: your bath was well below the temperature at which vegetables soften at all. Not slow — stopped.
So vegetables want 83 to 87 degrees, which is above the temperature you would ever cook a steak at, and roughly the temperature at which starch also gelatinises properly in potatoes.
Which answers your last question directly: no, not in one bath at one temperature. The two requirements do not overlap and no compromise temperature exists — anything hot enough for the carrots is well past well-done for the steak.
The good news is that vegetables at that temperature are quick. Carrots in something like an hour, depending on thickness. It is a sequencing problem, not an impossible one.