It's a fair perspective, but can I add another. Having worked professionally with both pro football players (at three Premiership clubs) and pro cyclists (world champion and Olympic level), when it comes to stress, burnout and mental health, the latter have it easy, trust me. Most of us can run around a field kicking, hitting or throwing a ball, but most of us can't ride a bike up an Alp, so we tend to make assumptions about the relative stresses of each on those bases. But burnout, albeit underpinned by biological processes, is largely psychological in origin. Very few pro cyclists will experience it in their careers, whilst many team sports players - soccer, hockey, cricket - will. Irrespective of the rights and wrongs at societal level, the media and business pressure on pro team sport athletes is a league above that on pro endurance athletes. So Southgate, rightly or wrongly in terms of how things should be, is correct in terms of how things are.
And yes, the cause of this is in many respects the hysteria you mention. But it's not just sport, it's music (burnout is common among successful and aspiring performers in all genres of music from rap to orchestral), and many other areas in which the media can build or shatter a person within seconds.