That really takes a dedication and long-term view that doesn't seem present with most orgs. A non-long-term view also impacts a lot of the actual software too - I'm working with a large enterprise company right now that has rather immense challenges - UX is so low down on the list right now. And yet... they keep getting new customers, and, from what I've seen, it's because their UX is still better than that of their competition, which is truly ... shocking.
But maybe it was a bit of a bubble in a sense. Neither I nor the other designers I knew would take a job at a place that didn't understand an appreciate the value of UX (unless that job was specifically to help change that), so that might be it. We didn't just want to be a mockup monkey spitting out pretty designs.
Frankly, I agree with you. The designers should have a stake in it. At the last place I worked, all designers received a daily report of costumer feedback gathered through the site. We attended monthly reviews of feedback and CSAT and summaries of customer care cases from the month, highlighting top trends and issues that the support staff were handling. We even had representatives from the support team attend our daily standups to raise issues and report back on our progress handling previous issues to the support team so they could be informed when customers asked about things. We even had goals around CSAT that impacted our bonuses. I used to read every single comment we received and even took the time to test and narrow down issues before working with a dev to find the root cause. We fixed some very nasty, but unusual bugs that way.
It comes down to the org: how mature is it with regards to CX and is everyone willing to make the effort to make it a priority?