Removing dead material and extraneous growth isn't something I do because it's a particular time of year, but whenever such circumstance is noticed. Removing dead material would never affect the bloom, dead branches don't bloom.
If something is distracting from an appearance that is pleasing to me, I don't wait to remove it, I want to enjoy the appearance today. Even if it's something that is about to bloom, doing that wouldn't affect the enjoyment of the blooms but enhance it, according to what my eyes want to see. Being distracted by an ugly branch, even a blooming one, doesn't please me.
Most advice about annual pruning is given with the assumption that a shrub must be pruned annually for some other reason than the spending of time and vigorous exercise, like to maintain height or girth restrictions, or are being sheared as a hedge, meatball, topiary, or lollipop.
Something like a Hydrangea, that retains dead branches, I like to trim when it's just starting to leaf out, when I can see which branches are dead. And on the live branches, which is the most vigorous node within the shape and size I want to see from it when it's in full glory. However, Hydrangeas that grow with no cultivation at all look gorgeous too. The micro-management of spring pruning is all for the aesthetic during the time before full new foliage & blooms.
You'll get to know your shrubs and develop your taste and style for shaping and maintaining them.
