I have had a salt system for 10 years now. If I were told I couldn't have salt system, I wouldn't have a pool. It is that much easier to maintain, plus tha water feels so much better.
I've had and seen no problems with salt damaging equipment. There are five pools in my cul-de-sac, all were built at the same time ten years ago, three were built as salt and one converted to salt. Two have had to replace their heaters, but one was the only no salt pool. Four have had to replace pumps, but again one was the non salt. So much for equipment damage.
As to salt damage to stone, there is no question there. In a hot, dry climate salt will erode and damage stone. If you were in Florida the almost daily afternoon rains would wash salt from the deck. But in AZ it is an issue. My pool was designed to be a salt pool, so no stone anywhere. My deck is cantilevered edge concrete coated with textured acrylic. I have one neighbor with a section of flagstone in the deck of their (converted) salt pool. There seems to be some flaking and erosion, and it is a bit more extensive than the non salt pool.
I don't have any experiance with salt an travertine, but from what I've seen on the forum tells me it's much harder and more durable than flagstone and that it can be sealed to slow (but not completely prevent) damage from salt. So, yes the better solution is change the deck material. If neither are an option, how to chlorinate the pool.
If you spend some time reading Pool School (the button on the top right of this page) you will learn that chlorine tabs, aka pucks are not an option. They are a recipe for disaster, particularly in a hot dry climate. Any solid form of chlorine brings with it something that will create a problem. Tabs add CYA, which builds up until it becomes necessary to drain the pool to lower it. So you are left with two choices SWG (salt) or liquid chlorine. You can manually dose the pool with,liquid or use a liquid feeder, either a Liquidator (gravity feed) or a stener pump to control and add chlorine.
Spend some time reading on this forum and you will find lots of folks who have tab feeders or UV/ozone devices that they no longer use. What I do know is that you live in AZ and want a Trouble Free Pool you can't use tabs as a primary source of chlorine.