Digital joysticks requires software support (as you have indicated with the installation of the drivers that you received on the driver CD).
As such, most digital joysticks cannot not be used with DOS games as most DOS software and gameports were built completely around an analog design.

MechWarrior 2 was probably the only major exception, as this game did support the digital functions of the Microsoft Sidewinder 3D Pro.
This particular joystick, although digital, has an "analog emulation mode", that also enables me to play games under DOS.

I'm not at all familiar with this Logitech joystick but, unless it has an "analog emulation mode" or similar, chances are that you will not be able to use it in DOS mode (even subsequent releases of the Microsoft Sidewinder joystick series had no direct DOS support).
You should, however, be able to use your joystick by playing your DOS games within a DOS window (command prompt), since it will then still be using the Windows drivers. However, there is no guarantee that it will work 100 %, since the game might not "respond" well to being played from within Windows.

Yeah - I was hoping this stick might have some sort of analogue mode too, but it has no obvious switches on it to enable such a thing and I haven't found any specific documentation for the stick (the manual the comes with the Wingman software is just a brief thing general to all Logitech game peripherals).

I'm sure I can get it to work with almost everything, it's just I have a few old flight simulators which I need to run in "proper" DOS mode as I use SetMul to step down my K6-III CPU to a 386/486 level so they run at the correct speed.

I also like to play some old flight simulators. Hence, why I held onto my Sidewinder 3D Pro for that purpose.
Not the most ergonomic of joysticks, but its analog emulation mode works quite good.
I use the later model Sidewinder joysticks for playing under Windows and the Sidewinder 3D Pro for DOS.

However, there were some good analog joystick manufacturers in the hey day of DOS and you might still be able to pick up an old one for a reasonable price.
The other users on here should be able to provide more advice on what models are good and which ones are very bad.