So, nobody is maintaining the roads in that neighborhood? It makes very little sense. The roads in my neighborhood are owned and maintained by the neighborhood (not that state or county) and we (the HOA) carry liability insurance.
My little 15-unit private drive was a single homeowner's 25-acre plot in 1950. Someone built a house in the back, with a long driveway up to it. Then they sold three lots for others to build nearby, then in the 1970s they sold another dozen lots, including mine. In the 80s they pooled some money together and turned the gravel drive into a paved road, still private, we all just have an easement that allows us to drive on it. Now, 40 years later, that paved road needs repaving to the tune of $90,000, because tree roots are breaking it up and it's delaminating in spots from an inadequate repair job a decade ago.
But try convincing 10 retirees with limited funds, and 5 families with small children, to each dig in their pockets and come up with $6,000 for a new driveway. Most people have bigger priorities, which is why there's currently a few nasty potholes on the route Google will suggest for you (the only route to my house).
This is even worse, because a bridge like that probably costs a lot more than $90,000, and instead of 14 out of 15 people driving over it every day, most of the neighborhood residents don't need it.
It still blows my mind that a bridge on an otherwise publicly accessible road was privately owned and maintained. That's a new one to me - around here, there are plenty of private neighborhood roads, but none have bridges.
Living with the graffiti seems like a strictly lesser problem than living with a road ending in a missing bridge.
They budget about $50/year to replace the signs. Still much less, even over rather a long period, than the cost of rebuilding a bridge.
Who actually owns the main road in your situation? As you each have easements, it sounds like the original house still owns it? I've heard of similar situations and it always confuses me, for just the reason you mention - I would never buy a house in neighborhood with no means to maintain then neighborhood infrastructure.
FWIW, I tend to agree that HOAs are less than ideal (and downright evil in some cases). But, I don't have a better solution the problem of shared common areas and infrastructure. I'd much prefer the county/state took over all roads, but in my neighborhood, we still have to deal with trash collection and common area maintenance (parking spaces, neighborhood entrance, signage, lighting on sidewalks, etc).
They can, but in my experience, anything that approaches a collapsed bridge in severity is resolved quickly. Large potholes can hang around, which can be unsafe, but anything bigger doesn't get left to fester.
Presumably a bridge being swept away in floods could be something insurance would weasel out of paying, as an "act of god" or whatever, so unless the HOA has enough reserve to cover the (potentially very large) expense of replacement, it comes down to the members approving an assessment to fix it. If the bridge being out didn't significantly affect the daily lives of the residents of the neighborhood, it kind of makes sense that they'd be loath to shell out who-knows-how-much on repair, on top of their regular dues.
This is one of the concepts addressed by StrongTowns.org, basically that American suburban infrastructure development has been following an unsustainable pattern for decades and we're now starting to reap what we've sown, here's an article on this exact topic: https://www.strongtowns.org/journal/2015/11/1/suburban-bailo...
Generally that equipment is useful for more than just paving roads and is owned by private contractors in the US - the same contractors could be hired by county, city, or neighborhood.
Sometimes they have too much work and contract some jobs to private paving contractors. The roads they build are flat, smooth, and drain well.
I'd prefer road maintenance was handled by the state, but the HOA works for maintaining common space and other services not handled by the government (trash collection).
If nobody in the neighbourhood can pay for a paved road, then maybe the road should be gravel. Especially a cul-de-sac -- the road exclusively serves the residents. If they won't pay for it, why should anyone else?