I think that part of the problem is that GPs say to go to A&E in three circumstances:
a) it is a emergency that they can diagnose, and it needs urgent treatment;
b) they have some concerns and know that A&E can do further tests that they cannot do, which could be helpful; or
c) they are out of their depth, and want to kick it higher up the chain, so nobody can accuse them of negligence.
The reason I know a lot about c) is because I have a very special needs child. Which is why, if a GP said to me about my normal children, you should go to A&E, I would most likely follow their advice. But I don't necessarily for my special needs child, as I can practically guarantee that I will be told to take him to A&E, in any circumstances, because his underlying condition is just too scary for any GP (the best you will get is they will ask me what I think). I therefore have learnt only to use the GP for him when I need something (like antibiotics). For my normal children, getting antibiotics is like pulling blood from a stone. For my special needs child, I have been offered antibiotics at the drop of a hat (including various situations where it is left to my discretion as to whether I should start them or not).
We had various similar incidents with one of the school nurses at his primary school. Always rushing him off to A&E as that was the safe thing to do and they didn't want ultimate responsibility. Meant that hours and hours of our time was spent running through his medical history with clueless registrars, who had never seen anything like him, in order to haul him out of there.
I that has meant that I have spent many many hours explaining to health professionals (and yes, safeguarding panels at one point) that given that he has a life limiting condition, there has to be a balance between his quality of life and his medical care. And while it would indeed be safer if he lived constantly in A&E, so that if he had a medical emergency somebody could just push a button and the best medical treatment could be offered, that outcome is truly awful for his quality of life. But yes, at least once I was reported to safeguarding by a paramedic who insisted on taking him to A&E when I told him that what he was seeing happened just about every month, and that A&E was not necessary, as he would recover on his own (I think paramedic couldn't take the fact that I laughed when he said he needed to go in on the blues and straight into resuss). Of course the flipside of this was that his care is and was so, so expensive, that nobody actually could face the cost of taking him away, as there was no way he could be fostered (parents are allowed to be on duty 24/7, but you can't place a child like that in foster care), so there was a bit of a dilemma there (and then when he turned 18, which nobody predicted would happen, we had a huge fight to actually find somewhere that was able to care for him as an adult). And the proof is somewhat in the pudding, in that he has now well and truly outlived his life expectancy, despite my thwarting his going to A&E on a regular basis (sometimes we did indeed take him, and sometimes we had to accept that school would send him, because it was just too scary for them, and that then the registrar would admit him, and it would then take days to get him out again). And of course we did take him regularly when he was very small (DH used to say that he had the frequent flier ticket into A&E) before we built up the level of expertise, by being his parents and watching him constantly, of deciding that his best interest was not always served by going.
So I certainly agree that medical professionals out there sometimes use A&E as a way of deflecting responsibility, or when they are out of their depth, and that a parent may sometimes know better, and that A&E is a horrible environment to take a sick child. On the other hand 16 months is not very old, and once is not very often in the scheme of things. From somebody who has spent months of their life in A&E with their child, often waiting until the early hours of the morning to be discharged or for a bed to become available for my child, two or three horrible nights is really not very often. If we were talking about five, 10 or twenty trips, and it was the same result, I think it would be more reasonable to think again.