There's some odd commentary going on here.
Doctors do double appts for one thing, and if you are so stressed you burst into tears talking about your work situation clearly you're distressed by it. No doctor can 'see' whether you are losing sleep, for instance, but they listen to people in pain and suffering all day, pretty sure they can see when someone is really suffering and unless you're an actor, well....
Doctors can see the difference between someone thriving and on top of their game and someone looking stressed and overtired.
Also, I've not come across an employer that pays full pay for limitless duration of sick absence! Some only pay ssp from day 1 of sickness, and offer nothing more. Obviously often the more senior the role and greater length of service increases sick pay benefits from company.
Hope work load and management can be adjusted for any returning to work! But, yes, horribly the higher you get it seems the less tolerance, very cut-throat it can be.