It’s something you can get pretty good at with practice, especially with a canoe. Honestly, it’s a whole lot harder with a kayak and in spite of our local club having more than one-hundred kayakers I’ve yet to meet one who is able to get through an obstacle like this without help, yet in a canoe this one would be not too hard. Maybe someday add a solo canoe to your fleet? (We need younger folks like you to keep that side of the paddling sport alive).
This is a case where painter lines usually become a necessity, because putting the boat into a tight place where you can climb out is often also a place from which it is impossible to extricate the boat just by lifting. In this case, you’ll probably have branches overlapping the top of the boat from any place you can park and also climb out, like next to that cut-off tree trunk at water level on the left, but if once you are out you give the boat a shove away from the tangle while holding the painter, you can pull it back and hoist the nose (via the painter line) over whatever tangles are present. You might commonly need to reverse that procedure on the other side, sliding the boat over some tangles that are too skinny to stand on, giving the boat a final shove to clear everything, and just as your boat is yelling “yippee, I’m free!” you can say “not so fast” and tug it closer again to where you can guide it underneath those same tangles the boat just went over, close in to where you can step in. Of course with a canoe, you can step in or out from the ends, if the ends are the only part of the boat that can be positioned close to a location within the snag having solid footing, though I avoid that when I can because that’s a much less stable situation. On that note, except in very rocky rivers, the water WILL be much deeper (emphasis on “much”) at these snag locations because of the way the current is deflected downward (even if that deflection only happens at higher water levels, the erosion of the river bottom has already happened by the time you get there).
I have often pulled a solo canoe, even with a load, over a fallen tree about the size of the one blocking that channel, but in this case I’d try to get my feet onto that bigger stump on the right.
As to the DNR having some responsibility to clear this stuff, I don’t believe that’s the case, and most places with blockages are far more remote, to the point that no one will be checking in the first place.