closer view

all over the rocks and "riverbed"

How deep is your soil? When you use a screwdriver and push it in the ground how far does it go?

Before working on the lawn/weeds we need to work on the soil depth and texture first.

I see multiple problems - first of all, in Phoenix, that seed mix they sold you is not going to survive and choke out the Bermuda, and it's not especially drought tolerant until it's been growing a couple of years. To change lawn types you have to kill the Bermuda first, which takes all of one summer, then plant the new species. And keep out the Bermuda that birds spread when they poop on the lawn.

You have the typical patch of grass surrounded by rocks ... it never works well because the grass spreads into the rocks and the rocks get kicked into the grass.

My recommendation is to get rid of the grass completely (unless there is an HOA rule that prevents it), install a drip irrigation system, and go for a xeriscape landscaping with colorful drought-tolerant shrubs and perennials. They do require maintenance, but it's mostly cooler weather tasks.

Scalping it down to dirt does nothing to kill the bermuda - its roots are several feet deep in your lawn. It will sprout back in the summer rains.

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You can spray all the clove oil and vinegar concoctions you want - it doesn't work on Bermuda unless you use 20% agricultural vinegar undiluted and that's nasty stuff and still doesn't kill the roots. You have to use something that kills the roots.

You can spread corn gluten and the ants and birds will thank you for the high protein snacks. It doesn't work here because it need moist soil so the inhibiting chemical can spread out into the dirt. By the time that happens, it's been eaten.

Even the synthetic pre-emergents are mostly useless in Phoenix. They need to be in the top 1 inch or so of soil to kill the emerging seedlings If it rains too heavily, it washes it down too deep. If it doesn't rain, the seeds just wait until next year to sprout: either way it's a wasted effort.

If you spray now with glyphosate (NOT the current Roundup formula, which is mixed with another weed killer, get the pure generic concentrate glyphosate from WalMart: called Eliminator Weed & Grass Killer) you will kill off the spring annual weeds before they set more seeds.

You can try "solarizing" them, by putting a layer of clear plastic over the top fopr a week or two but that does nothing to kill the Bermuda.

In May, when the Bermuda is out of dormancy and growing well, you can start killing it: it's an all-summer project. (see the link)

While you are killing the Bermuda you can be planning what to plant next fall for the shrubs and perennials.

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Used according to the package, handling glyphosate is safer than handling clove oil and definitely safer than agricultural vinegar.

As for the cost ... if you buy the concentrate and follow the dilution recommendations, it's cheaper than vinegar per gallon.

In AZ, because of our high pH water, it increasesa the effectiveness to add a tablespoon of vinegar per gallon to lower the pH, and add a few drops of hand dish soap as a surfactant to get through the fuzzy leaves of most of our weeds.

NOTE: I'm a certified desert landscaper. We studied the local problems thoroughly, especially the "what to do with a lously lawn" problem.

Here is a link that might be useful: How to Kill Bermuda Grass