You won't be unkillable, and if the enemy built offense, you'll be just as even in trades. The damage is so small that it won't matter a whole lot if the person wearing the vest isn't already tanky. Moreover, selling the Bramble Vest after laning phase will put you extremely behind in gold.
In order to get away with this, and come out actually ahead, that Bramble Vest will need to get you 3 kills or 900g, to make up for the fact that you delayed your actual offensive item purchases and will be down in gold.
Because Bramble Vest offers no offensive stats, it's a terrible bargain to think you can get kills with it as a class focused around building damage. Moreover, Grievous Wounds is useless during laning phase except maybe in niche cases VS a Soraka. But even then, the GW only lasts 1 second so they can disengage and heal up.
By this logic, ADCs should invest into Warden's Mail because it has 40 Armor and reduces the other ADC's attack speed by 15%, allowing you to attack faster than them and come ahead in trades.