It does no harm to the plant to cut the blooms. Many people remove them and bring them inside to make flower arrangements.

I sometimes get all stems killed, except for one survivor that remains from last year. Because it is older, it is longer than the new growth. It does not look good because it is out of sync with the rest. Usually when this happens, the stem survives but the flower buds do not survive so, for aesthetic reasons then, I cut it off so it is in sync with the other stems.

The stems with no blooms may or may not produce blooms. I am unsure if the snow cover protected last year's stems or -my guess- if these smaller stems are new growth from this year. We will have to ask the hydrangea! Ha! But my point is that, if any of those stems without blooms had a flower bud, cutting it off now -just when they normally bloom from new wood- may cause you to loose some bloomage. But if this possible loss is not important, you could prune thestems with no blooms too.