Take a look at the A-10 (Warthog or Thunderbird) as an ideal surface attack air platform.

Sure, it can carry missiles and bombs, but it's primary reason to exist is to act as an aerial platform for a big Avenger Cannon. This is basically a massive machine gun capable of cutting down infantry, vehicles, some buildings and tanks, etc.

The cannon was designed specifically to solve certain military 'problems'. Then, the military handed the cannon to an aircraft designer and said 'build a plane around that'.

So they did.

**Addressing technology edit in OP**  
The avenger cannon came about after the 50s to be sure; but AFAIK there's nothing in that cannon which isn't **possible** in the 50s. It's just a large machine gun. Sure, there are some alloys that could make it lighter, but if you want a weapon for ground attacks, and you're dealing with 50s tech, you're going to end up with a cannon of this kind eventually. It might be heavier and have some other problems (like needing a cooling system) but as we discuss later, you don't have missiles on the A-10 in your world so we can afford a little more weight. **end edit**

It's not an air combat superiority fighter. Pit one of these against (say) an F-18 Hornet, and the Hornet will win hands down. That said, it's fast, has a wing design that can support a massive amount of damage and still be capable of flight, and can bring a massive amount of firepower to bear on ground based targets - missiles and bombs were pretty much added to the payload as an afterthought because with wings like these, why not load them up? If you don't have that extra weight though, perhaps there's more room for ammo for the cannon.

These kinds of planes are the new cavalry.  Instead of horses charging an infantry line, causing massive havoc and then getting out, these planes fly over, strafe the enemy (causing massive havoc) then fly away before AA weapons can be brought to bear.

Why are they dropping in military usage today? Drones. If you look at a reaper or predator drone, you can almost see the A-10 DNA in its shape. Remote ground based strategic strikes are now taking the place of the more conventional battlefield strafing runs because surface attacks are now more surgical in nature and designed to have a large impact on enemy capability or strategy; almost a long range sniper role. Now that modern warfare isn't against state actors, the specific niche of the A-10 is facing extinction.

That said, in your world, it seems like the perfect plane for this kind of activity. It also fits with your battleship bias. Both are weapons that are effectively platforms for guns. The A-10 is just air based instead of water based. Battleships will still have strong AA capabilities, meaning that without bombs, A-10s are almost useless against heavily armed ships capable of defending themselves from airborne attack. On the ground however, they still make a lot of sense for traditional, battlefield based combat.