Not Mowing Bluegrass
Jefferson County Colorado
Yes, in short, this can work well. Many golf courses have done this - for many years now. It is something that I have recommended for golf courses and HOAs as a "trial run" before they do the bluegrass to native (buffalograss or blue grama) conversion - which you have correctly pointed out as being expensive. The trial run with keeping the bluegrass but not mowing it allows people to see what a native conversion MIGHT look like. If it's not desirable for many or most of the interested parties, you can always go back to mowing the bluegrass as before.
The bluegrass may not "stay green" through the summer without mowing and water. Under dry conditions, the bluegrass will go dormant - and turn brown. But, in some summers, it just might retain a good level of green without supplemental irrigation.
If you do decide to try this, I would NOT fertilize the area. In fact, these greenbelt areas - even when mowed - often require little or no fertilization (maybe once, in early fall). In my experience, these green belt areas - besides being overwatered - are often overmanaged (too much fertilizer, unnecessary herbicide use).
I would encourage you to try this on a small area - perhaps one irrigation station - before stopping maintenance and irrigation on all of your bluegrass green belt areas. Further, if you have trees growing in the areas where you propose the irrigation reduction, a plan will have to be made for watering the trees. The biggest problem I see when HOAs convert to native (or stop watering their bluegrass) is that they don't realize that the trees growing in these areas were also being watered by the "turf irrigation". Trees can rapidly decline when they don't receive regular irrigation, depending on the species of tree. This is a common misconception about trees in Colorado - that they can exist and grow well without supplemental water. The trees we have growing in our cities, parks, HOAs, yards, etc. weren't here before the Front Range was settled. They are here because we created irrigated landscapes.
Let me know if you have other questions or if I have forgotten to answer anything.