Pressing the thumbstick does indeed switch it to an analog mode (that red led indeed indicating which mode it is in). Which, I suppose, means that technically there would be four additional "buttons" on that stick (not usable at the same time as the analog functionality, of course).
I figured that if I use the three buttons on the base as combo buttons with the central hat of the stick, it kind of resembles the configuration of a gamepad, where the three base buttons act like the X, Y and B face buttons of the gamepad, and the central hat acts as a d-pad (after all, it even looks a bit like a d-pad), which somewhat allows transferring muscle memory. (Still need to figure out what to use as "shoulder buttons", but I suppose the secondary index finger trigger can do that work, as it has two buttons to it.)
That being said, moving from having used a game controller for two years (and for a couple of decades with other games before that) to using the joystick feels a bit daunting at first. While in principle the controls resemble those of the game controller, it's still completely different in terms of muscle memory.
There's also the thing that a game controller just acts almost perfectly as a universal controller regardless of where you are in the game. In menus, in the different in-ship panels, in FSS mode, in the system map, in the galaxy map, flying the ship, controlling the external camera, driving the SRV, and even on foot, a game controller (especially a modern one like the Xbox Series X controller) has been polished and perfected to be fluent in any use, even in completely different situations (like navigating UI menus, driving a vehicle, or even walking in first-person perspective). Of course it's not absolutely perfect in every possible situation (most prominently in first-person shooting because of the limited speed and accuracy of aiming), but it's very usable and fluent even in those less-than-perfect modes, especially when one has 20+ years of experience in using a game controller precisely for such forms of control. In other words, the game controller is perfectly suitable and adapts to any part of the game, any mode, any screen, any form of control, seamlessly, easily and effortlessly, and even the default button mappings for the most part are very good for all of those modes (requiring only a few tweaks). I don't need to switch controllers if I eg. fly a ship and then go on foot, or start driving an SRV, or start navigating the game's menus.
The joystick, on the other hand, doesn't feel as universal. It feels significantly more specific and less suitable for some modes. Navigating menus is passable, but a bit awkward. Driving the SRV is less than ideal (a game controller, in contrast, being pretty much designed for such a thing), and it's very unsuitable for on-foot content.
Of course this is just talking from very little experience on actually using the joystick in practice, but still...
(Also, I wish the game supported dual settings for everything, so that I wouldn't have to choose between the joystick or the game controller in the control settings. It does support this for most if not all button presses, but it doesn't seem to support this for analog sticks and triggers. For some reason only one setting is provided for these. This means I can't just change between the joystick and the game controller if I wanted to, as it would require going through the analog stick settings and change them.)