Besides the "physical" constraints of launch windows (orbital parameters, technical limits like fueling), keeping a launch window "open" is also associated with a large fiscal and organizational expense. There are many things that "need to happen" for a launch window to be realized, such as:
- Road closures
- NOTAM + Air traffic monitoring
- Launch corridor ocean clearance
- and more...
These are all not cheap, and it is directly possible to put a dollar-per-minute figure on them. For example, to ensure that the ocean is clear in the keep-out-zone, someone has to be paid to go out and actually do this. Communicate with traffic, utilize radar or similar systems to ensure it's clear, and if necessary, send coast guard military ships out to police the waves (the dollar-per-minute operating costs of military vessels are not cheap).
A similar situation is applicable with air traffic. The launch window and keep out zone needs to be communicated to all aviators and Air Traffic Controllers so that they can re-route aircraft out of this area. This, again, is connected to an actual financial cost. Airlines might incur financial damage by obeying a keep out zone (eg. they burn extra fuel) and someone has to communicate with all the ATC folks to make sure nothing goes wrong.
Even for the road closures point, some government or government-authorized official has to go out and put physical signs on the roads to stop traffic. In some cases, just like with air traffic, this disturbs commercial activity where shipping needs to re-route or delivery services suddenly can't reach their destinations.
The bottom line is that keeping a launch window open means imposing a small inconvenience on many people, and while a lot of this is covered under "good will" or "taxpayer dollars" along with the actual money that SpaceX pays for the launch licenses, the people calling the shots only have a certain tolerance limit.
I don't know specifically where the 20 minute launch window for Starship IFT2 came from, but it was not related to orbital mechanics or the technical limitations of the rocket and launch procedure. Therefore, yes, the launch window's shortness was administrative or bureaucratic in nature.