I used to get very hung up about working out every day too (well, every second day in my case, I used to alternate between workouts and rest days). I don't think there's anything I can say that will convince you to not worry about it because I don't think I could have been convinced in the past, it's just something I had to learn from experience.
Something that helped me loosen the reins a little though was changing it from "no missing a workout" to "can't skip workouts two days a row". That way if it got too late to do my workout and I hadn't done it yet the consequences weren't so drastic. Rather than just giving up on the entire streak because of a missed day, it became "okay do it tomorrow instead, set an alarm so you don't forget".
My way of getting around the need to have a streak nowadays is just to lower the bar ridiculously low. I feel like I've shown up for the day even if I just do the EotD, which takes like 30 seconds, or my physio exercises.
Another idea! (Sorry this is becoming an awfully long reply.) Do you have a rule against finishing the program regardless of days off and THEN starting again with an attempt to do it every day? So rather than going back to the start of the program whenever you miss a day, you can say "alright this run-through is stuffed, let's finish it off and then try again" and each time you finish the program and start again you could try and beat your consecutive days record? And the fact you've got to finish the program before you can start again ups the stakes a bit.
This idea's a bit counter-intuitive, because it seems like raising the pressure/consequences of missing a workout is going in the complete opposite direction - making the streak a bigger deal when logically you know it shouldn't be a big deal at all - but what ends up happening is 1. you are actively practicing not being a perfectionist every time you finish the program imperfectly and 2. as you said, you'll be better off than when you started!
I don't know if any of this is helpful or it's just rambling - this is what I would've told me from five years ago.