I am struggling to see how arcane gate can be used in any but the most specific circumstances.

It is generally quoted as being good for things like an invasion force getting through city walls or a mass prison break but there is a limitation that you have to be able to see both points and they must be on the ground. 

An invasion force can get through a portcullis using this spell but not city walls / gates that the caster can not see both sides of.  Trying to land on the Ramparts wouldn't work because even if the DM classed the top of the ramparts as ground the caster would only be able to see the wall not the ground. One point being within 10 feet of the caster also rules out flying up to a point he can see both sides of the city wall. 

Prison escapes would have similar issues so unless the enemy have been kind enough to provide a visual on your desired destination I can't see a way of using it. 

The only other thing I have come up with is if a large group need to cross a natural barrier such as a chasm or fast flowing river but 6th level seems very high for something like that.

Any other ways it can be used?  Or are there other ways round the restrictions for invasions / prison breaks?

The best part of Arcane Gate is that either portal can be rotated with a bonus action.

You put your exit portal on top of a cliff, or at the edge of a volcano, a vat of acid, whatever... the exit points safely away from the hazard. You lure the BBEG into the entry portal, running like crazy, (dash action), then use your bonus action to turn the exit portal around 180 degrees. When the BBEG runs through after you, he gets dumped straight off the cliff/into the volcano/whatever. It's a bit of a "Saturday Morning" gag, but it's effective.

As written, the two arcane gates are indistinguishable except one starts no more than 10' from the caster. You can call one the exit, but stuff can go both ways.

The caster and others can go through and come back. Useful if you are porting a dragon horde from a cave to the surface.

The best use is to have an impenetrable chamber more than 10' diameter with 2 slightly smaller openings, preferably one over the other or side by side. Go in, cast Arcane Gate at the openings with the mist facing out. Those outside cannot enter or see in. Their stuff will go back at them. Those within can see out, and their spells and attacks will have full effect without even splashing back.

I think you need to think about more than just moving people.  The bottom of a river could be ground you could see and within 10'.  Or use it to precision aim siege weapons.  Or stick your hands through and cast cone of cold.

Honestly, 500ft sight range teleport the entire party is pretty good on its own.  There are ways to get elevation while staying on the ground.  Any being able to keep it open just opens up more possibilities.

And imagine the hijinks that could ensue with readying your action to cast the spell

Teleportation magic was extremely strong in old editions, but WotC decided that they didn't like the fact that parties basically stopped walking anywhere once they hit double digits.  So every edition since 3.0 has hit teleportation spells with the Nerf bat.

Order of Scribes can make decent use of this using Manifest Mind to see farther away and cast the gates at greater distance. The object you see through is intangible and doesn't occupy space - meaning it can go through walls. 

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Another use is if you're in a narrow corridor being chased you can make portals on either end to trap pursuers . 

If you know the enemy fires ranged things, you can ready the spell to cast when they are about to fire to redirect their attack.

.. There's quite a lot you can do, if you're creative enough.

The narrow corridor needs to accommodate 10' rings. The spells fails if it overlaps a creature, but overlapping inanimate stuff is up to the GM. I would not allow the rings to intersect walls and ceilings.

The spell opens above points on the ground. There will be gaps unless the floor is a 5' concave radius.

Quote from JimGettman >>

The narrow corridor needs to accommodate 10' rings. The spells fails if it overlaps a creature, but overlapping inanimate stuff is up to the GM. I would not allow the rings to intersect walls and ceilings.

The spell opens above points on the ground. There will be gaps unless the floor is a 5' concave radius.

 
The "gap" isn't specified by the spell beyond saying "inches", so unless the pursuer can squeeze through gaps 2 inches wide, it can still work depending on the hallway size.

How would you rule Arcane Eye, Ghostly Gaze or Scrying with Arcane Gate?  Think all 3 would work?

Casting Arcane Eye requires concentration, you can see a far away location.  You begin concentrating on Arcane Gate and hop through.  Ghostly Gaze and Scrying similarly have this concentration mechanic but I would allow all of these uses.

Murder the big bad and then don't bother going in the dungeon.

How about in a 10 foot hallway you cast arcane gate on both sides of someone so their only option is to walk in an endless loop that you can't escape unless you dispel magic or can teleport or something.  Poor casters force cage.  

You could close a passageway with it.  The first ring in front of you, facing you.  The second 20ft down the passage facing away from you.  Once your party is through, reverse the second ring to face the other way and essentially close the path.

You could use it as a perfect shield against projectiles, and deposit those projectiles somewhere useful.

As long as one side is against a wall you can always open/close the one side as needed, making it useful for stopping multiple enemies from getting through.

Arcane Eye to get into the inner sanctum of a dungeon and avoid all the traps.

What makes spells like this valuable is their multi-functionality.  Kind of why I always take things like Telekinesis, Bigby’s Hand, Animate Objects, etc - you can always find a use of these spells, in combat or otherwise. 

Send your spider familiar into the castle while you hide outside near a river of flooded moat.

cast the spell, one opening under water, and the other one in the throne room, flooding the castle.

As a high lvl spell, I would give you leeway for creative use.

Please ask your DM for permission first and don’t spring it on him in the middle of the session.

Wow, All I need now is a 10 foot pit of acid and that gets a lot more scary.  Thanks for the hot tip.

Two points, on the ground, that you can see. Perpendicular - which basically just means 'they're upright'. Most tricks don't work - but if you happen to be fighting on a volcano for some reason, the OP's does. Although fighting 500' from the open caldera of a volcano that's currently erupting might ... a bit of a stretch. All tropes aside.

One day (or the same day, if you have two 6th level spell slots) use Move Earth to create a hill tall enough to let you see over the wall, with a ramp on the side opposite the wall to let you (and your army) reach the top. This gives you a nice observation / command post. [While doing so you could dig a trench in front of the hill, to make it more difficult if their army wants to try to reach you.] Line up your army on the ramp then the next time you have a second 6th level slot, cast Arcane Gate from the top of the hill. Put the first portal 10 feet below the hill, on the ramp (which is made up of earth and so I'd argue is part of "the ground"), with the second portal inside the enemy's city. As soon as the portal appears, your army starts marching through.

You could even do this if you don't have two 6th level spell slots or don't know both spells. You could use a spell scroll of Move Earth. I looked to see if there was another magic item that could cast it, but the only one I found offhand was Lyre of Building. I suppose if you have a bard friend they could build you the observation post.