I have the following sentence: 你的表有一点儿快。I understand this sentence to have the meaning in English of "Your watch is a little bit fast." But if I were put this sentence into Chinese, I would say "你的表很快"， or "你的表一点儿快". Why is there 有 the first sentence, and what is its meaning/grammatical purpose?
你的表 (your watch) is the topic
有一点儿快 ( is a little bit fast) is the comment
in '有一点儿快', '有' (having) indicates 'possession' of a condiction or ability, which is 'being faster than normal'
therefore, the literal translation of the sentence would be 'your watch is having a problem of being a little bit faster than normal'
[(topic) + (comment)] sentence structure is different from typical S+V+O structure like "你的表(subject) 行(verb) 快了 (adverb)"
Since 你的表有一点儿快 is a [(topic) + (comment)] structure, '一点儿快' describes something the watch possesses, therefore you need '有' to indicate this possession
I think you can consider 有一点儿 and 一点儿 as different phrases, because their usage are different.
有一点儿, meaning 'a bit', is put before the word that it modifies. As in your example: 你的表有一点儿快. Another example is: 这问题有一点儿难（this question is a bit hard）.
一点儿 is put after the word as a complement, denoting 'extent' or 'degree' enhanced a bit. For example: 把题目出得难一点儿. (Make the questions a bit harder).
Compare 有一点儿难 and 难一点儿: 有一点儿难 describe that 'something is being a bit hard to deal with'; 难一点儿 is used to describe 'to increase the degree of difficulty a bit'
Your sentence 你的表有一点儿快 means the time indicated by your watch is a bit ahead to the actual time. Say your watch shows it's 6:15, but the actual time is 6:00. If you want to use 一点儿, you might have to put it after 快: 快一点儿. 一点儿快 doesn't make sense in any case I can think of. Well, 快一点儿 means "go faster a bit" or just "go faster", which is quite different from 有一点儿快. So, 快一点儿 doesn't fit for your sentence, because it means differently.
Note: there is another uasage of 一点儿. It can be used to emphasize the negative. For example, 你说的一点都没错！meaning you are absolutely right(without even a subtle mistake). This usage is different from what you are asking.
Hope this could help you out.