On a leverage bit (one with shanks), a cub strap is always necessary. If you've seen someone ride without it on a curb bit, they were mis-imformed and using the bit wrong.
With snaffles, the chin strap is a matter of preference. It is not necessary, but can be used to keep the bit-ring from being pulled through the mouth if for some reason one rein was pulled that hard.
In your situation, with your french link snaffle, you don't need the chin strap. It shouldn't come in to play at all. Since your horse was acting different without it on, maybe you had it adjusted too tight in the past? Here is a picture of a properly fitted chin strap: Chin strap on snaffle image by janlvtt on Photobucket
See how it doesn't touch the horse's chin? It would only do anything if the rider say, fell and held on to one rein.
Also, make certain that, like in the picture, the chin strap is in front of the reins on the bit.