Exclusive Common Area--gate to subdivision

We are receiving some pushback on using association funds to install an automatic iron gate at the entryway to one of the cul-de-sac subdivisions (FLorida community). Members with homes not on the cul-de-sac maintain that their money should not go to a gate they will never get/have no reason to use. The issue lies in that the roads are all common areas for which all owners share an easement, so we may need to give all owners the gate code in order to not obstruct the easement to the common road. The funds remain an additional issue--has anyone dealt with this matter, know any case law or pertinent 720 statutes or posts that may speak to whether owners can object to payment for common elements that do not benefit them individually in any way? Other than blocking the road, I would think the gate would be similar to a sign or mailbox cluster that only benefits one portion of the entire community, but everyone must pay for.

Let us spend association money to build/maintain a pool that only people on a specific street in the association can use. This makes as much sense as what the OP enquired about.