We have all been pushed to the limit at times.
Your son will have learned an uncomfortable life lesson that even those who love us dearly have their limits. The situation is not ideal of course, but the lesson that you can actually push people too far is one that he will need in his future life.
Things often go like this when a new baby is in the house - he is saying "Look at me! I matter too!" So it is probably best not to say that you cannot do something he wants because you have to do something for the baby. He will gradually get used to that idea, but it is a difficult lesson.
Don't beat yourself up. Take it as an opportunity to try and find ways of dealing with his defiant behaviour that do not escalate as things did for you today.
I agree with previous posters. Scrap the morning shower; scrap insisting he eats breakfast - he will find out that he gets hungry without it; get him dressed when he first gets up - help him, but praise him when he does some of it himself. If he fails to finish up dressed in time then take him in his night clothes. If he chucks his spoon across the room, he has to get it back himself or get a new one or do without breakfast - obviously not ideal, but sometimes we just have to be practical.
The morning routine is hell with more than one child - I had 3 at 3 different schools at one point and you could bet your life one would suddenly decide they needed their PE kit (or cookery ingredients, or a model of a dinosaur that was supposed to have been made last night or ..... any number of things!) as soon as we were all settled in the car!
I would suggest that you stop any more apologies - one is enough. If you keep referring back to it, it could easily become a stick to beat you with. Honestly, you have said sorry; he gets it.
The first thing you need to do is forgive yourself, put it behind you and move on. If you get drowned in guilt it will influence how you handle him - you will be tempted to spoil him as a penance rather than being the adult rock who puts firm boundaries in place.
There will be a lot more things you will do as a parent over the years that will not be ideal - but children need to grasp that life is not ideal; that they have to find ways to accept that or they will be very miserable.
When my children used to say "It's not fair", I (very boringly) used to say that life is often not fair, but we have to accept that and move on.
Forgive yourself, move on, enjoy your children - when they are not driving you mad!! 