I sometimes wonder whether this announcement had any connection to the introduction of Class 456s on SWT/SWR, seeing as a passcom activation in the rear portion of non-gangwayed stock could be a headache if the Guard has moved into the front portion of the train for any reason.
This is about delays, rather than about the welfare of an ill passenger, although of course it has to be packaged in a customer-friendly way. If somebody is unwell to a significant degree, alerting the Guard at the earliest opportunity is the best course of action. This allows assistance to be arranged and coordinated so as to be waiting for the train at the optimal location, and getting the train to that location more quickly if need be. This may involve missing stops, making extra stops or various other measures to expedite the process of getting help to someone who needs it. Staggering onto a platform in the middle of the New Forest with "excuse me, I think I had a heart attack ten minutes ago" isn't very helpful. If operating a passcom is the best or only way of contacting on board staff in such a situation, then there should be no issue with that at all. A life or death medical situation is very definitely an emergency. GSMR is available where mobile phone reception isn't, and that's a good few areas of the railway, and only the crew have access to it.
I really don't think that many people are sufficiently stupid to pull a passcom because they have a headache or a runny nose, are they?