I'm no garden or lawn expert but I consider Bermuda more weed than a desirable plant. It's really invasive in places you don't want it (like flower beds and vegetable gardens adjacent to the lawn) and tough as nails to get out. It makes for a durable lawn, once established, but looks half dead in the winter. Not really familiar with zoysia. It may provide green during cool weather but it won't keep Bermuda stolons from running rampant.
This very well can be the case, depending on your location. Where I am located, we can go one of two directions. Either warm season grasses such as Bermuda, Zoysia, or Centipede, or cool season such as Fescue or Bluegrass. In my area, it's about a 50/50 split, dictated by amount of shade.
While the cool season grasses do stay green year round, they are also very delicate, and basically have to be babied during the warmer months. Water requirements are high, as is instances of crippling fungus. Also, you have to overseed yearly maintain a healthy stand. They are also not ideal for heavy foot/pet traffic.
With Bermuda & Zoysia, you do have a dormant period each year, which can be supplemented with annual ryegrass very inexpensively. The dormant period can actually work in your favor, because you can use inexpensive non-selective herbicides (glyphosophate/2-4-d mixed) in february/early march to kill the rye, along with any weeds. 2-3 weeks later and it looks like a switch was hit, and everything turns bright green. This can be greatly amplified by giving a healthy dose of potassium just prior to the grass going dormant, and a good shot of iron as it is coming out of dormancy. Keeping it out of beds pretty much has to be done chemically with glyphosphate (roundup), which fortunately is very cheap these days. The key is buying the generic variety, and making sure that it is not labeled as being "quick kill" or "fast acting." The versions that advertise as such
Again, this is all specific to location. An hour away from here where the altitude is significantly higher, and the seasonal high temps are a bit more moderate, fescue is king, and super easy to keep looking good.