Looking at the images you've attached, this appears to be an 8 ft wide by 6 ft tall fence. That's a good amount of weight in just the wood, and there isn't any part of the design that diagonally braces the frame, except the steel cable... which tore from its mounts.
My layman's view is that you absolutely need diagonal wood elements, which should only be installed after unloading the fence, either by removing the boards or by propping up the wheel-end so the frame returns to being squared. If the wheel interferes with this, remove it for the time being.
But I think you'd still need the steel cable, and if that has broken from its originally designed mooring, then this gate is already compromised. You may have to start over with a new Adjust-A-Gate kit or repair the current one so the cable will mount to the steel parts, rather than the wood.
I would say to rectify the diagonal supports first, before doing anything with the hinges, since if the hinges were actually the root problem, this gate would have already fallen over. That said, it seems to me that such a wide gate might have called for more substantial hinges.
The other commenter's suggestion to consider a pair of less-wide gates is also sound, if the goal is a minimal-fuss gate that will last at least a decade of additional sagging and weather.
From an urban planning perspective, there are some caveats to your points:
Cut-and-cover will make shallow underground tunnels cheaper to construct in almost all cases irrespective of building in an old city center or as part of building a new city center from scratch. In fact, older pre-WW2 cities are almost ideal for cut-and-cover because the tunnels can follow the street grid, yielding a tunnel which will be near to already-built destinations, while minimizing costly curves.
Probably the worst scenario for cut-and-cover is when the surface street has unnecessary curves and detours (eg American suburban arterials). So either the tunnel follows the curve and becomes weirdly farther from major destinations, or it's built in segments using cut-and-cover where possible and digging for the rest.
At least in America, where agricultural land at the edges of metropolitan areas is still cheap, the last 70 years do not suggest huge roads, huge offices, and huge house lead to a utopia. Instead, we just get car-dependency and sprawl, as well as dead shopping malls. The benefits of this accrued to the prior generations, who wheeled-and-dealed in speculative suburban house flipping, and saddled cities with sprawling infrastructure that the existing tax base cannot afford.
It is, until it isn't. Greenfield development "would be short term appealing but still expensive when it comes to building everything". It's a rare case in America where post-WW2 greenfield housing or commercial developments pay sufficient tax to maintain the municipal services those developments require.
Look at any one municipal utility and it becomes apparent that the costs scale by length or area, but the revenue scales by businesses/households. The math doesn't suggest we need Singapore-levels of density, but constant sprawling expansion will put American cities on the brink of bankruptcy. As it stands, regressive property tax policies result in dense neighborhoods subsidizing sprawling neighborhood, but with nothing in return except more traffic and wastewater.
Either these cities must be permitted to somehow break away from their failed and costly suburban experiments, or the costs must be internalized upon greenfield development, which might not make it cheap anymore.