Why does the caramel sauce fail

Walk away.

No, really - it works. Once you add the water to the sugar, it will likely harden and clump up (I don't know how to stop that happening). But once it does, you can just walk away, and leave it to sit till it cools down.

A lot of the sugar will just dissolve on its own, given time and enough water to dissolve into. Some mixing (occasional stirs as or after it cools) will dissolve some more. And at the last, when most of it has dissolved, you can gently start heating the pan up again, stir it around, maybe bring it to a boil for a bit, maybe add a bit more water and heat some more to get at those last undissolved chunks...

It's sugar, in water. It will dissolve until the water's saturated. It takes time to dissolve on its own, and that's annoying if you want to use it right away - hence measures like heating, or stirring, or crushing the dried caramel to stir into water that way. But it really works just as well, and is less tedious, just to give it some of that time and let it dissolve away.

I did this when making a caramelized burnt-sugar syrup, and it worked. Adding the water was pretty spluttery at first (the caramelization had to be stopped pretty fast), but I did notice the caramel hardening to the bottom of the pot once it had settled a bit. I gave it a few stirs, and walked away to take care of something else - and by the time I got back to it, most of the lumps were well on their way to dissolving.

Caramelising is a chemical process in which sugars decompose under the influence of heat (pyrolisis). It happens to any heated sugars, no matter if they are free (as in heating refined sugar for making candy) or bound in something else (such as the sugars naturally occurring in an onion). The outcome of the process are compounds which have a dark color and pleasant aroma.

Sauteeing is a cooking technique. It consists of frying small pieces of solid food on very high temperature with very little fat while shaking the pan all the time, so they won't overheat and/or stick.

Many people don't know the technical meaning of sauteeing and use the word for plain shallow frying at medium temperature and without shaking the pan. This seems to be the definition your employer was using.

In fact, you were shallow frying both kinds of vegetables, which resulted in caramelisation plus other changes for the onions and in these other changes only for the vegetables which don't contain significant amounts of sugar.

Cooking recipes frequently avoid saying just "fry the onions", because onions have to be brought to a different state for different recipes. So they usually use a word which implies a desired final state, such as "caramelize". For many other vegetables, which only have a single usable state of doneness, they specify the technique instead. This is why different words can be used for the same process - one describes the technique you are using, the other describes the changes which are happening, it is like saying 'I am going sunbathing' or 'I am going to catch some tan', which are indeed the same process.

Incidentally, true sauteeing is a bad idea if you want to caramelise onions. Low and slow is the way to go if you want caramelised onions, while during the high temperatures used in sauteeing they go from translucent to burnt without passing a nice caramelized stage. But just leaving them in the pan for some time is a good technique.

You also say grilling. This is a completely different technique, and it is done on a grill instead of in a pan. I can't imagine how you would caramelize cut onions on a grill, unless you put a griddle on the grill, which is equivalent to frying.