I recently snapped my 2-piece wooden Greenland paddle while practicing my rolls in shallow water. The break happened near the ferule, and it looked like this:
The problem, with any repair is that to make it as strong as the parent wood, the repair is going to have to fit inside. So the dowel or rod used must be even stronger then that parent wood. In doing that you need something denser and such things are also heavier then the original wood. When it’s done the balance is going to shift towards the repaired side. In the case of that one you show, the break is close to the center so the shift of balance is not going to be extreme. If you were to use Osage or something similar and get about 3" of penetration into each end you may find the paddle is nearly as good as new. But the farther out from center you get the more extreme the balance shift becomes. To make it as strong as it was (or stronger) you need to use Stainless Steel as the dowel rod and you can make it super strong, but the offset in balance is extreme if you use metal.
I also have broken a few paddles. Because I make them myself I have not tried to repair any of my own. My repairs have all been for friends and customers on their paddles.
I use Kajak Sport Splits on my 2 piece paddles so I just made a new half for the two of my own which I broke. See here>>> Greenland Paddle Split - Kajak Sport
When the break is like this one, I just make a new paddle --------------or new half of a paddle.
Were you pushing off the bottom? I’ve only seen broken paddles when the blade hit a rock or a reef while the kayak was getting maytagged. If you were pushing off the bottom you were lucky the paddle broke and you didn’t damage your shoulder. With the paddle in an extended position and trying to force the boat over it’s pretty common to end up with a shoulder injury.
I make wooden bows for a hobby and a break like that looks like there was rotten wood ( seepage a round the ferrule perhaps). If you bought the paddle I would contact the seller to discuss what’s going on with the wood. Most stress fractures look more like the photo posted by Szihn above.