Branches off the trunk need to be no more than about a third the diameter of trunk at point of the branches attachment. Your branches are much thicker than that and may require much more room to become fruitful as a result, depending on variety..

I would rebuild the whole tree by removing all four scaffolds and use new growth to shape a better tree or cut off the top and use the lower two branches to create an apple fence.

It 's all about leaving the least vigorous shoots and removing the most vigorous from a scaffold when training an espalier (and often, free standing trees as well).

So much easier to train if you start with a dwarfing rootstock.

Many varieties bare only on two or three year old shoots that haven't been cut back. If you do cut back shoots you need to do it 2 or 3 times during summer and this will sometimes coax the tree into forming flower buds on the shoots while keeping the shoots compact.

There are lots of other tricks you can look into like scoring the trunk shortly after trees first show growth (used often by commercial growers) or root pruning but I think your problem lies in excessive pruning and oversized scaffolds..

With fruit trees, you need to take a long-term point of view. What I would do now, or certainly next spring, is to take some of those upright watersprouts near the trunk and bend them to the horizontal. A convenient way to do this would be to tie them to the large horizontal scaffolds. This will induce them to flower and fruit the year after next. Don't leave any growing vertical. Either bend them or prune them out. Wherever you want a new scaffold to grow, bend a watersprout. After these have established, you can cut back those long lower horizontal scaffolds that aren't fruiting.

I see no reason why those upper branches shouldn't be fruiting now if the tree is healthy. But the fact that one branch has died suggests it isn't.

Avoid fertilizing it. This will just give you more vegetative growth and less fruit.

For pruning, go buy a GOOD pair of bypass hand pruners, loppers, and an actual pruning saw.

Your tree was certainly planted and trained when young as an espalier, to grow against that fence. Looks like you have two pairs of scaffolds - the larger on the bottom and smaller on the top. I can't see quite clearly, but it looks like the upper right-hand scaffold branch has died completely. You have a choice - to keep it that way or let it grow away from the fence into the yard.

Heed what Harvestman says. He does this for a living.

I would make 4 or 5 cuts a half inch apart with a sharp saw a third of the way into the wood of your new leader starting at the base of the crotch on the right side (the side facing the current leader). Then carefully bend the left branch to straighten it and tape it to the remaining wood on the right with electric tape. You can use two of the small branches on the top of the new leader to be the start of your 2nd tier.

If the wood will bend without making a hinge you can skip that part but it looks like it is probably too rigid to bend without it.

Incidentally, I know it is hard to separate expert from not so expert advice on the internet, but for the last 25 years I've spent about 180 work days a year doing nothing but this kind of technical fruit tree pruning.

You can Google Alan Haigh, fruit trees, NY and probably find some insurance that I know what I'm talking about. If I'm going to spend all this time writing directions I don't want it to be wasted, although that's always the risk.

Wouldn't that be the 3rd tier. It looks like there is a tier about 1 foot off the ground, maybe 20-25% the height of the tree.

To recap, clean off all the deadwood from the dead branch (but don't cut it off). Then make cuts into the left branch, enough to bend it straight upward, and then tape it to the right branch... then allow one of the small branches to become the new "right-side" branch, and cut off the other one(s).

For all the other big shoots that are growing upward, try to bend them sideways so they grow sideways out (vs up). I used to cut off all shoots that were growing up (which meant, basically all shoots!), but it seems that's not the right approach.

I should make a video of making a hinge to bend a branch but don't have the equipment. The important thing is not to cut (much) more than a third of the diameter and do it on the side your are bending it. Use as many cuts as it takes to make the branch reasonably limber working your way up the branch.

The suggestion of taking it out of espalier form and turning into a conventional free standing tree is also a workable strategy and good idea, but the question is, would this work within the space? Espaliers are attractive but somewhat complicated to maintain as productive trees. But then, that can be said for all corrective fruit tree pruning.

Apple trees are relatively forgiving of transplantation.

Here is a publication that begins to explain some of the biological issues you are dealing with. If you are interested, I can find you more.