Looks like a Dodonaea viscosa doing poorly.

The Dodonaea viscosa has flowers but my plant has never had any flowers.

Usually, only happy plants flower. That plant looks very unhappy. And it may not be Dodonaea viscosa, but that's what it looks liek to me.

It, also, resembles Acacia longifolia.

My plant is definitely not an Acacia longifolia either (mine used to have long medium-green leaves with no flowers).

I can't id it but it certainly looks miserable. The leaves are chlorotic and it has been badly hacked about. As RugbyHukr says if it isn't happy it won't flower.

I prune it several times a year to keep it the same size as adjacent shrubs. The plant went chlorotic since last October and I've removed a lot of the dead leaves.

I'd remove it and plant something meant to stay the size you want.

What would've caused the chlorosis so quickly in this plant? (The shrubs - although different species - on both sides of this plant are heathly).

Will applying chelated iron, manganese sulfate, iron sulfate and/or organic compost/mulch bring this plant back to health?

I 'd have to go along with Ruby and suggest it could be more like a form or variation of A. longifolia than a Dodonea.

The cutting back would be enough to prevent flowering - there are no shrubs (that aren't effectively sterile variants) that don't flower when big or old enough. Flowers not having been seen on a particular example is not an automatic indication that a suggested identification is incorrect.