I don’t doubt that you speak French. French people tend to have difficulty distinguishing those sounds because they are not distinguished in French. In English, they are: English has a much larger inventory of distinct vowel sounds than French (or indeed most European languages). In typical French-accented English, “bit” is indeed pronounced like the French word “bite”, but in native speaker English, it is not.

I am a native speaker of American English and also speak French quite well. If you neither accept personal experience, nor what is written on Wiktionary, what evidence would you accept?