>still the guilt made me feel like a failure and increased my anxiety when I'm on-call.
I don't know if this will help with your mindset at all, but as someone who's been listed as the secondary if the first responder doesn't answer: I really wouldn't beat yourself up over it. Self-correction is fine, but try not to let it drag you down too far. We have the same worries too. Our worries are usually "Crud. If I don't pick up, then it's either going to (big boss) or the customer is going to be firing off nasty emails in the morning and dragging me into some stupid meeting." The cause of why we're being paged doesn't really come into play. At most, I might just send a text or whatever to the first responder, to make sure that they're okay. "What if they aren't responding because they were in a car accident?"
If you have a good manager - and it sounds like you do, since you weren't reprimanded for performing a fundemental human function - then they are probably doing their best to look out for you and your interests. When people talk about "not being the boss" or "working as a team" or whatever, generally what they're trying to get at is that "They have your back". They've been in your position and have probably felt that same crush of emotion and worry that you might deal with. They are meant to be the final filter, in a hierarchy of filters, to protect you from those outside elements which ruin your ability to work effectively.
If you put in a sincere effort at your job, you have nothing to feel guilty over. That might be easier said than done, but it really is true. It just might take some time and practice to be able to forgive yourself when you make a mistake.
And if you feel like you could be doing a better job at this, or at that, then you just need to permit yourself the time to improve on that thing, and to remind yourself that you are allowed to mess up from time to time. It's about how you learn from those things.