If “engineers” in the title of your post means “software engineers” (aka the developers), then I'm sorry to say that you are wrong:

A software engineer opted for this career path because we want to code, not pick up calls or tickets.

Even with the best intentions when suggesting this, you must understand we never wanted this role. That's why we are not “support engineers” (as you refer to in your first paragraph) but “software engineers”, and that's why we have different names: because we focus on different aspects of the process.

Give a support role to a programmer, and they will quit or at least lose motivation for that position. Developers like to code, not attend support requests.