I often have my snowshoes on my back while ascending a broken popular trail, and I hate it! I'll wear my 36" Tubbs on a hard-packed trail if I possibly can because I like the traction and I don't like them on my back. The slightest slip of the toe drains energy. I wear shoes on Tuckerman Ravine Trail.
If a trail is broken out with the little MSR type shoes, then only those shoes fit into the track. So it's microspikes for the climb and then off-trailing for the way down on snowshoes that float on deep snow.
I have some 13x28 Army surplus bearpaws which I used for most of my first round of the winter NH then-46. Great for breaking trail uphill in deep snow, not so good breaking downhill as the flat toes tend to dig in causing face plants and somersaults. I haven't worn them in years as most steep trails are broken far too narrow for them and the wide flat trails don't need that much flotation.
If you have 10x36 Green Mountain bearpaws (once the most common size on AMC hikes), they aren't bad for snowshoeing but are a giant pain to carry. The weight is only a minor part of the story, but imagine tying them crosswise on the top of your pack - fine for the Wilderness Trail but on a typical trail they will brush on overhanging tree branches pulling you off balance and dumping snow down your neck. If you mount them vertically high on your pack they have the same problem while lower down they may swing into the back of your leg. In the wind above treeline they act like a sail - annoying and perhaps even dangerous. So it's hardly a surprise that owners of such snowshoes might choose to leave them in the car if they think they'll carry them all day.
But with the smaller size of snowshoes popular today, there's much less excuse not to bring them. My two smallest pairs of snowshoes will tie easily on a pack and will even fit inside the pack I usually carry in winter (ultimate anti-snag location). Sure, people trying to set speed records need to save every ounce and it's a pain for those who don't own snowshoes to rent something they don't expect to use, but it's hard to see why so many people still leave snowshoes in the car for thieves to get