As you already said:
Spinach is my least favourite vegetable.
There's nothing wrong with that and it's perfectly understandable.
On the other hand:
Spinach is the vegetable I don't like the most.
Even though this is understandable, it does not sound natural simply because nobody uses that phrasing.
A more natural version would be:
Spinach is the vegetable I dislike the most.
In this particular sentence construction, dislike is preferable to don't like. (Although don't like can be used in other constructions without a problem.)
The reason behind this is that when we use a comparative term, it's in conjunction with a single word rather than with a word and its negation.
These sound natural:
The most likely.
The greatest strength.
The least weak.
These alternate versions sound strange:
The least not unlikely.
The greatest not weakness.
The least not strong.