This certainly can work, especially in the earliest days, when you have very product-minded and entrepreneurial engineers. But once you start growing past that founding team, you probably end up with engineers who will burn out if they have to do this, or otherwise you are limiting your engineering talent pool to people who are excited (or tolerant) to work this way. It also becomes a problem when the engineering org begins to split into teams and individual engineers no longer work with the entire customer experience.

I would suggest thinking of the results you want to achieve and figuring out what's the most efficient way of getting there. If it's understanding the customer pain points, can you do it by shadowing support periodically, in lieu of actually doing support?