Well, if you don’t put enough helium in, it doesn’t.
It’s not like the stuff’s magic and put any amount into a balloon and it will float.
What you have to do is put enough of it in to counteract the weight you mentioned.
Think of it this way. If helium gas was visible and you just opened up the valve on that helium tank, you’d see all that gas heading straight up. Now all by itself those helium molecules have weight, so why are they going up?
Because air has more weight/density than helium does. So helium floats upwards till it reaches a balance point between gravity’s pull on its weight and the density of the other gases around it.
Ok… back to the balloon. You’ve captured this gas in an envelope of mylar, this ads considerable weight to the same volume of helium, but if you put enough helium into that envelope, it still wants to float upwards because the weight of the balloon is nothing near the weight of the big old tank of gas that you filled the balloon from. So if allowed to, the balloon would float up to a similar (if much lower due to the weight of the mylar) balance point.
Think of this in a similar manner to ordinary air in water. If you take that same mylar envelope, does it float if it’s totally empty? I’m not sure, maybe it settles to the bottom of the tub, but take a deep breath and exhale into that mylar envelope, and it will float right up to the surface.
(I’m sure someone more versed in chemistry & physics will clarify the slight muddle I’ve made of density & weight)