Lots of problems here. The most important is the one you've picked on, and yes, your explanation is on the right lines. The usual explanation is that there are 3 types of adjectives - scale ( or gradable) and extreme adjectives and limit adjectives. Gradable adjectives , such as hot can be "more or less" - something can be anywhere along a scale including freezing, cold, warm, hot, boiling .... The ones at each end are the extreme adjectives of the scale. Scale adjectives can be modified by expressions like very, terribly or a bit when they're in the middle of the scale very hot, a bit cold, etc. This is logical because those expressions themselve express degree and simply move the adjective up or down a bit along the scale. The extreme adjectives, on the other hand, can't be moved, they are absolutes and require absolute modifiers like absolutely eg - absolutely freezing. Similarly, the adjectives in the middle of the scale can form the comparative and superlative - colder, coldest - while the extreme adjectives can't *more/most boiling.
Limit adjectives can't be conected to a scale - they are either/or, eg dead, alive, identical, true, false. These are complicated - we sometimes treat them as gradable - very true - and sometimes as limit - completely true, sometimes can't modify them at all - *very/totally dead - and sometimes can only do so metaphorically : I felt totally alive.
So what about lovely ? Something can't be very lovely or slightly lovely so it doesn't seem to be in the middle of a scale (though why not? Isn't gorgeous further up?). We'd use limit modifiers - absolutely lovely. But it can be used in the comparative/superlative - lovelier, loveliest -so it seems that it is scalar after all!
The modifier that springs to mind as collocating best with lovly is really - weird in itself, because it can be used with all sorts of adjectives - really cold, really freezing, really identical.
And because it is so universally applicable, that's the one I tell my students to use if they're not sure. Adjective modification is a horribly complicated area - apart from the factors discussed above, there's collocation (which words go with which) to think about too - why is something horribly complicated and not *strongly complicated for instance? So you need to be able to simplify it for them and provide "safe" paths through the jungle.
Notice that the rest of the sentence also sounds weird. Firstly, strictly speaking family being a singular countable noun should not be used with all, but with the whole of. However, as the family is a singular unit composed of a plural number of people, it's often treated as plural, but then the verb becomes plural too. So I'd accept any of the following :
My family is really lovely.
My family are really lovely.
All the people in my family are really lovely.
The whole of my family is really lovely.
? All my family are really lovely.
The last one seems odd to me, but I could imagine a native speaker saying it, whereas your student's version seems less likely. It's not nearly such an important mistake as very lovely though.
I'd be interested to know what your tutors say when you discuss this.