The second 3-way switch has the common wire from the 14/3 attached, and then the other screws are occupied with the hot and neutral coming from the fixtures.
You are not describing it correctly.
1. The neutral wires are not involved in the switching.
The red wire from the 14/3 goes into a wire nut with hot wires that come from the power source and go to the next switch on the circuit
Here is how a 3way circuit is wired.
The common terminal on one 3W sw is wired to a hot wire (constant 120v)
The common terminal of the other switch is wired to the switch leg (wire going to the light)
There are two wires that connect the other two terminals of both switches (travelers)
It sounds like they used the
red to send the power upstairs to the first switch and
black and white as travelers.
The black wire going to the fixtures is connected to the common on the bottom of the stairs.
There shoud be two whites connected together in the upstairs box (source neutral and lights neutral.
it sounds like I can leave the existing three way switches in their current locations by removing the end of the 14/3 from the bottom of the stairs
Correct. pull it out and drop in to one 4way location, then the next, then back to the downstairs 3W.
IF the upstairs switch has the RED on the common, wirenut the reds together in the 4way boxes and use the black/whites as travelers.
When you install the 4way switches, I don't remember if they go side to side or tops/bottoms on the switch (it will have 4 terminals. Look at the diagram on the box.