I don't think that this is a very good strategy. It can only lead to reduced benefit of the index. The single user that searches for "caramel" will not find that single résumé that includes that word. But you have only made a miniscule reduction of the index size.

The performance should not really matter whether the stopwords are there or not. First, I don't think many of these 10000 words are frequent enough to reduce the size. But more importantly, an index is an index, which means that the size of the index does not matter much when you seek it, since you follow the B-tree (or whatever organisation a fulltext index has.)