A mophead hydrangea in Zone 5 would not bloom reliably unless it was a rebloomer variety or unless you winter protected it. Amongst the rebloomers, the list of whites is limited. Two that I recall are the Forever & Ever Series’ Peace and the Endless Summer Blushing Bride (BB). There may be more; if someone remembers, please chime in. This last one I mentioned, BB, is actually a very, very light pink; almost white. Amongst the mopheads that only blooms white and that will need winter protection are Mme. Emille Mouilliere, Sister Theresa; Wedding Gown (reblooms), etc. This last one also has a nice fall foliage and is a hybrid (part mophead and part lacecap). I do not have it but I was told that the leaves turn wine red/purple-ish in the Fall. Check which one you like best based on criteria that you like.
Hydrangea arborescens, aka Smooth Hydrangea, is more winter hardy in Zone 5 than mopheads/lacecaps and should also be considered as it blooms on new wood, so it does not typically need winter protection. It can get 5’ high/wide in southern gardens and can spread thru suckers. There are new Invincibelle varieties hitting stores too. Invincibelle Ruby, Spirit and Spirit II are less tall than Hydrangea Annabelle's 5-6' height and all three bloom pink, just different shades of pink. Mini Mauvette is a new Invincibelle, even shorter and, of course, has a different shade of pink. Invincibelle Blush has a “normal” height but its blooms are light pink. Invincibelle Wee White is a new, more compact, white bloomer while Invincibelle Limetta has lime-colored blooms. New too, Lime Rickey has slightly different lime colored blooms. Sounds confusing at first but, I asked the wholesaler and they said that Lime Rickey (not called Invincibelle Lime Rickey; just Lime Rickey only) is much larger than Invincibelle Limetta: 4-5' tall versus 2.5-3' for Limetta. The flowers of Lime Rickey are much greener and the blooms are larger and have a kind of flatter shape, while the flowers of Limetta start green, go white, then go back to green and are also much more ball-like. Both have good strong stems, but Lime Rickey has extremely strong, rigid stems. This not only ensures the flowers stay upright all season, it also gives the plant a stronger, bolder presence in the landscape. Invincibelle Limetta reblooms, producing new flowers all season long, and while Lime Rickey remains showy all season, it does not continue to produce fresh blooms. All of these are quite winter hardy and do not need winter protection in your zone. In southern gardens, it needs afternoon shade but it may do fine in full sun in the northern/cold areas of the country. I have seen pictures of a hedge of these arborescens shrubs where the person planted white/pink/white/pink/etc and it looked very nice.
Hydrangea paniculata is yet another choice but carefully check the height for those that you may be considering as most paniculatas, including Limelight, tend to be quite tall, well beyond your specs. The smallest one that I recall is probably Bobo, which may get about 4.5’ high. Little Lime, Strawberry Sundae and Little Quickfire are slightly taller, around 5’ for LL/SS and 6’ for LQF. Personal note... Limelight blooms -to me- appear to be like pure white but LL seems to be off white. Not sure why that should be or why I noticed that but oh well. Just a fyi in case you want (or do not want) certain shades of white. Paniculatas are extremely winter hardy too. They, as well as the oakleaf hydrangeas, are probably the more sun tolerant of all hydrangeas. But down here, even they need afternoon shade; in northern/colder locations, they can take full sun.
Hydrangea quercifolia, aka oakleaf hydrangeas are also hardy to Zone 5 but choose short varieties. Like paniculatas, most varieties tend to be tall. Some short ones are Pee Wee (4’ tall), Hayes Starburst (2-4’) , Munchkin (3’ tall and 4.5’ wide), Ruby Slippers (3.5’ high, 5’ wide), Sykes Dwarf (2-3'), Vaughn's Lillie (around 4') and Jetstream (5-6'). Oakleaf hydrangeas have awesome foliage color in the Fall. The more sun they get, the better the Fall foliage and bloom colors. For some reason, my RS has had blooming issues on two years after it was planted. Then it finally bloomed on year three. It was odd because Pee Wee and Snow Queen had no issues; those two though are in a more protected location. We'll see if cold weather was a problem because this year we crashed down to 13F, which is unusually cold for the area. We'll see what all the oakleaves do in the Spring... blooming-wise.
Take some of those heights with a grain of salt. In southern gardens with longer growing seasons, these plants tend to grow taller. If you see ads where they specify a range (example: 5-6 feet), the shrub may be closer to 5’ in your shorter growing season.