(note: not trying to say the medical diagnosis scenario you describe is less important than your contribution to getting the world off SMS, just spitballing how we can work towards timely updates in our current world
and wean off SMS)
in the situation you describe, or any urgent situation where speed of communication is paramount, what about bombardment through multiple channels? like, i'll often leave my phone out of my pocket, and not pay super close attention to it. and if it lights up with one text message, or one signal message, or whatever, i might not look at it. but if it's buzzing like crazy, or someone starts calling, i'd pick it up.
i guess what i'm saying is, "urgent" to me means signal/text/call/call someone that might be around the person/whatever, until the message gets through. if something is urgent, i would not send it solely by text. i've certainly had SMS messages get dropped or delayed many many times over the years.
can you really only use one messaging app at a time? signal is my primary messaging app, but i don't really find it bothersome to use whatsapp and regular SMS also. different people i communicate with prefer different channels, and often the same person will use different channels with me depending on the purpose (e.g., my dad mostly chats with me by SMS, and most of my immediate family's group chat is on SMS, but when my dad is texting with me about some sensitive personal financial info, it's over signal).
also, i hope that whatever the urgent issue was, it was resolved in an ok way. like i said, not trying to shortchange the urgency of a medical emergency or second guess your decision making or frustration at the time.