I would want to add that it also depends on how you see a product manager! Is this is someone "only" doing user research and managing backlogs OR is this a former senior developer, who co-creates the product with the engineers, having the user's voice and an eye on business strategy. The latter one is hard to replace by a UX designer.
This brings me to the following: I would say you can maybe cut out the business guy playing PM (and usually then only doing project management) - since either the dev or design background person can also easily do his/her job together with either the dev or design part.
Personally, I usually bring a UX designer and technically skilled PM to the table. This works best at my playground. No need for a simple "project manager".