The answer to this is generally no. Particularly not when it comes to fluency. Unless you have some sort of an impairment, you already have enough fluency in your first language. Learning French or Latin will do nothing to improve it hinder it. However, learning Latin and/or French may increase your familiarity with the cultural and linguistic underpinnings of much of writing in English over the last several hundred years. So, you may in effect become a more fluent reader. It may also help you understanding distinctions that are hidden in modern English orthography - therefore, you may become a more accurate (and perhaps by extension fluent) speller.
However, this does not apply across the board. Learning old Germanic or even older varieties of English will not help you at all with any of the above. As won't learning old Church Slavonic to help you improve your Polish. English is quite a unique case here because language contact has played such a profound role for so long and in so many ways. While language contact has formed all known languages to some extent, I find it hard to imagine learning the source languages would be any use whatsoever in the way I outlined above in any other case of contact-induced change.
Also, I cannot stress enough, how marginal any potential improvements would be from learning Latin or French on your English. If you are concerned with your fluency, vocabulary or spelling in English, you would be far better served by learning more of those, rather than trying to fix the problem through another language.
But if you already know the other language (at least in part), you can certainly benefit from drawing on that knowledge. I found knowing English useful when learning French vocabulary and I've also come across a textbook of French from the 1930s based on knowledge of Latin. My (albeit rudimentary) knowledge of Latin has definitely been useful for dealing with English words of Latin origin. But this also comes with the danger of following the patterns of the original language be it with false friends, literally translated idioms or syntactic patterns. I would compare such knowledge to being the occasional useful stepping stone on the muddy journey through language learning but not anything resembling a stone walkway.
You also mention knowledge about language such as that by scholars. Every scholar of English should know enough about French, Latin or Old English to understand their impact on the current form, but unless they study historical developments of English, they only need to know about it rather than have any detailed knowledge of either of the source languages.