When a wheelbarrow is not a practical solution for everyday use, a sled is the answer.
I live in Nebraska where, if we don't have 6 inches of ice, snow or mud, then it must be 100 degrees Fahrenheit in the shade. With our weather, a wheelbarrow is not a practical solution for everyday use (also, the older I get, the less cooperative the darned wheelbarrow seems to be). Our solution is a sled. It pulls just fine over mud, grass and snow.
I use my cheap plastic sled year round and it makes moving hay bales quite easy. More than once, my wife simply plunked herself down on the bale in the sled and rode both down the hill to feed our horses.
Watch for winter closeouts on nice, big toboggans. Keep it away from the grandkids, and you should be able to make your sled last for seven or eight years.