From what I am aware of, I have known that CWA ballasts benefit HID lamps because such ballasts allow for better current regulation than simple series chokes and HX autotransformer ballasts, which allow for better lumen maintenance during brownouts and voltage surges. I wonder if fluorescent tubes benefit from being operated on CWA ballasts too.

Not much. Fluorescents are way more tolerant towards lower currents, they are able to restart instantly. So the supply brownouts are not that big deal as with HID, where any extinguish it means lamp becoming off for many minutes.
Plus the use case is different: HIDs are used on chain installations stretching long distances (streets, roads,...), where the voltage gradually sags as you are going further from the feed points. There is impractical to adjust the currents manually or rely on multiple ballast subtypes to account for the sag as that would be prone to installation errors.
On the contrary, fluorescents are used all fixtures crammed on a small area of a single building, where the feed voltage is way more uniform.
Plus another issue: The CWA has a rather high current crest factor. For rather heavy HID electrodes it is not that much of a problem because of their thermal inertia, but the thin fluorescent filaments would suffer from too high temperature ripple, so becoming too hot after the peak and too cold during the valey. Consequently the emission layer would get consumed very fast. Many lamps (designed for HF only) would suffer from this already with the series choke...

The lead lag ballasts are similar to CWA ballasts. So just fire up your good ol Tulamps!

The point of a CWA is to utilize the core saturation in order to stabilize the current. This saturation then, as a side effect, causes worse current crest factor.
The fluorescent lead-lag ballasts do not do that, they keep the inductor in its linear range.