I think it's an experience/confidence thing. I'm guessing this is your first child and you don't have a huge amount of experience with babies? If so perfectly fine and sensible to err on the side of caution and no hcp will say otherwise really
That said I personally wouldn't have taken to emergency for a slightly raised temp and increased breathing (that's normal for babies btw and even adults on occasion it's the body trying to cool itself) I'd have stripped baby to nappy, given calpol, a cool drink if they'll take one or increased bf if bf and cooled the room they were in slightly - the weather is getting warmer and it takes some babies a while to cope with that.
BUT I've a LOT of experience with babies inc sick ones. My own dd runs hot and in her early childhood had febrile convulsions. Very frightening but normal for her at this point. First time I did call ambulance as it was first one and a bad one, but after that and getting advice on prevention and treatment (a refresher/update for me really) I didn't always call someone if it was a fairly mild one but would get her checked at gp next morning as an urgent appointment
What you may find helpful - and frankly I think all new parents should do this if only to help them stay calm - is doing a baby 1st aid course. I'm out of touch with who does the best ones now and if they're available to do online with covid but the Red Cross, St. John ambulance, st Andrew's ambulance here in Scotland from memory were always pretty good on this stuff
Some links
https://www.redcross.org.uk/first-aid/learn-first-aid-for-babies-and-children##
https://www.sja.org.uk/get-advice/first-aid-advice/paediatric-first-aid/
Always better to learn this stuff BEFORE you need it if only so you're not stressed and panicking! Kids can be buggers for accidents/illness that scare the bejesus out of us 
My dd it much later turned out has a disability. It makes her more prone to accidents (she's had some SPECTACULAR ones over the years that could have been written for "some mothers do ave em"!), more susceptible to certain infections and it's why she tends to run hot too - so a high temp for others is normal for her, that took us a while to learn/get our heads around.
I'm sure she will be fine but it's hard not to worry 
(Doesn't get any easier as they get older dd is 20 and I worry more about her now!)