some way to slide a vertex along an edge and still snap it on one axis.

The workaround for this is to use a custom transform orientation and the 3D cursor. The process goes like this:

1. Select the edge that you want to move your vertex along.
2. Create a new custom transform orientation based on that selection (expand the Transform Orientations roll-out and click the Plus[+] icon in the top right corner of the roll-out).
3. Select your disconnected vertex
4. Move the 3D cursor to that vertex (<kbd>Shift</kbd>+<kbd>S</kbd>→Cursor to Selected).
5. Change your pivot point to be the 3D cursor (you can use the pivot point roll-out or the <kbd>.</kbd> hotkey)
6. Select the vertex you want to move.
7. Scale to 0 along the Y axis (<kbd>S</kbd>→<kbd>Y</kbd>→<kbd>0</kbd>→<kbd>Enter</kbd>... you could also use the Scale tool, but the hotkeys are more exact way of doing it).

It looks like a lot of steps, but I've broken it down to really small steps for clarity's sake. Doing it at speed isn't too troublesome, unless you have to do it a lot. Then it might be worth making an add-on that wraps all of these steps into a single operator.

blender.community/c/rightclickselect/SScbbc/

Have you tried selecting the vertice and press Shift+V?

Leaving a comment in case someone knows how to do this.