Join nodes is not playing nice

Okay, it's probably me.

I have two curves and want to join the end of one with mid-way along the other. Let's say I am trying to achieve something like a 'Y'. The left is angled moving to vertical down the page, the right is angled. There is a node at the point that the left changes from angled to vertical.

Firstly, I know I can't join midline, so I break the left curve at the node. I now have two line-end nodes on the left where once there was a regular node.

Secondly, I know that they need to be combined, so I select all three lines and combine them.

Now, I place all three line-ends on top of each other using Snap.

The last thing is to join the nodes into one. I have tried Join Curves and set the tolerance and radius to be larger than any gap (they are on the same point) and nothing happens

There is something missing in my understanding.

The reason I'm doing this is that I need to produce a cutting line for a vinyl plotter and it is quite a complex shape.