My absolute favorite tool for getting through blackberry canes is a Nokogama sickle, with a serrated curved blade. There doesn't seem to be a great English name for this tool - it's sometimes called a Japanese gardening knife, gardening sickle, Japanese sickle, or similar.
Getting through blackberries is still work - but this is the best way I've found, makes it easy to cut and move the canes one-handed. Best of all, the serrated blade can be used to grab and flick the canes without needing to grab them directly.
After starting to cut my way in, I'll usually step on canes, and slice them off at the bottom. If they're especially large, the tool can be used to cut them smaller or move them out of the way. As they're cut, they'll tend to lay down, and you can walk over them. A huge mass of blackberry can originate from just a few canes in a small clump. It's not fast, but not painfully slow either. In a half-hour, you can make it a few hundred feet or so.
I hesitate to bring this up since it's off-topic - but since nobody else has mentioned it - himalayan blackberry (Rubus armeniacus) is a nasty and common invasive plant in these parts. Volunteers spend thousands and thousands of hours grubbing it out and removing it as part of trail maintenance and land restoration. No land manager is going to cry over it, especially if you're brushing out an existing trail.
Taken literally, leave no trace principles say that a baby English holly sapling found in the woods should be left alone. In the Pacific Northwest, that's not good stewardship.