Nets Don't Do Damage
The reason you can't add damage to the net from Sharpshooter (or other feat) is that a net doesn't do damage in the first place. This can be seen in the entry for net on the Weapons table on page 149 of the PHB.
Damage for a net is undefined. If there is no damage type, no damage die, and no damage value in the first place, damage isn't a category of the effects of a net in combat. It's effect in combat is "special" and is listed on PHB p. 148 as somehting that, on a hit, will restrain a creature. Restraining by itself, as a Condition (see Appendix A, PHB, Conditions, p. 292) also does not apply damage.
You can also address this as a matter of specific over general. The specific exception to weapons doing damage is the net, as spelled out in the Special Weapons paragraph.
Net. A Large or smaller creature hit by a net is restrained until it is freed. A net has no effect on creatures that are formless, or creatures that are Huge or larger. A creature can use its action to make a DC 10 Strength check, freeing itself or another creature within its reach on a success. Dealing 5 slashing damage to the net (AC 10) also frees the creature without harming it, ending the effect and destroying the net. When you use an action, bonus action, or reaction to attack with a net, you can make only one attack regardless of the number of attacks you can normally make.
Put in a more mechanistic way: as there is no damage entry per the weapons table, there is nothing to add +10 to. You can make an attack. The effect of a successful attack isn't damage, it is a condition: restrained.
Look at the weapons table from top to bottom. Every weapon has a damage die (range of value) associated with it, and a damage type.
The entry for net has no damage value and no damage type. It doesn't do damage.
Damage by a net is something like dividing by zero: undefined.
Note: for those with older versions of the PHB, Unarmed strike was removed from the Weapons table in the PHB errata. Thanks to @Miniman