Tecomaria capensis, aka Cape honeysuckle. It can be grown as a vining plant, a sprawling shrub or trained into a more tidy, upright form. You might want to Google it to see how you would like yours to look. Agree that pruning into a large, round ball is not an ideal approach :-)

umm, not sure I'd pull it out, the hummingbirds love this bush. I leave mine pretty much un-pruned. Depends on where it's located if that solution would work for you. Can you add Texas Sage in another spot?

The round look is obtained by shearing. It's the cheapest way to maintain a landscape and therefore is used pretty much everywhere. Shearing stops many plants from flowering very well.

An alternative is called selective pruning. Just cut specific branches as needed to obtain the shape you want. Used almost only by DIY gardeners. Most "landscapers" are not gardeners and know almost nothing about plants and they don't need to.

And of course the third choice is no pruning.

I wouldn't "open it up" (at least not a lot). Many of plants here (in Phoenix) need their trunks shaded, at least here in Phoenix. This plant doesn't strictly required a shaded trunk to stay alive but I think they do better.

Many gardening techniques used other places do not work here. Like in most areas opening up this plant would be a good thing (more air flow) but not in Phoenix.

Saying you're in AZ doesn't really tell much so I'm just talking from a Phoenix perspective. Probably really bad advice if you're in another area.

This plant is frost / freeze sensitive, so you may want to wait until late February of next year to do much pruning. Although often listed as a shrub Cape Honeysuckle can be grown as a vine, if desireid.

Here is a link that might be useful: Cape Honeysuckle