I used to have permission to cut down and dead stuff if I cleaned up well and scattered the brush.
To do that I used a quad and atv trailer. Years later I added a LogRite arch.
The reason for getting the trailer first was that I could get trunk wood cut in rounds, and the smaller limb wood. In addition, the trailer was useful for many other tasks, where as the arch is a dedicated tool, of little use for limb wood, or anything else. It is fun to use though, and almost essential if your milling lumber.
The arch does have benefits. A lot of firewood can be moved easily in a shorter amount of time than cutting rounds and handling individually.
Together the arch and atv trailer were ideal where I was cutting, leaving little if any foot print in the woods, and clean logs for cutting later when time allowed.
Time constraints (along with tools available) may effect your "the quickest way", depending if you work in large blocks of time, such as weekends, or if you process firewood an hour here and there. Especially if there is travel time, set up time, etc. that may push you to focus on gathering first, and then processing. Or if time allows, combining the two.
A very affordable tool is a pulp hook. It is the cheapest, single most effective tool, for handling small rounds when gathering up firewood in the woods.
My own situation was to gather when I had bigger blocks of time and process little by little in the evenings, or longer when gathering was not an option.