As an Assistant AD who does mostly game management we offer both digital and paper tickets. We don't advertise that we offer paper tickets, let's just call it our "secret menu". In other words if you ask about it we offer it, if you don't ask we don't tell.
a.) digital tickets are more efficient, yes. I don't have to make a late night run to the bank to do the night deposit, we don't have to take as much time on a late Friday night doing the paper money count in a back room somewhere in the building. If done correctly it can expedite the process for getting people into the stadium. I got news for everyone, these things called Smart Phones ain't going away. This thing called the internet seems to have caught on to society. Paper tickets should be offered IMPO as an option but I find it funny that people are still pushing back against online ticket sales, especially when you consider that much of life is now and has been online. I know people in their 70s who have social media, do all their shopping on amazon but buy a ticket to a high school football game online!?!?!?!!? GET OFF MY LAWN!!!!! Someone explain this to me as I simply don't get it.
b.) I can't speak for every district but if my school wants new uniforms, to be able to turn on the lights, mow the fields, etc... then we need to sell tickets. My boss (our AD), one of his evaluation indicators that he has to discuss every 9 weeks with our principal and district admin is where are we / how are we doing financially. So to say the AD doesn't care about money would essentially get him reprimanded with a letter in his personnel file and then eventually termination if it continues to be a problem. That would be like saying a teacher doesn't care about taking attendance (state mandated btw). We don't have to necessarily be turning a profit but part of those meetings (I've sat in on them) does involve the question "well what are you doing to try to solve this problem?"
c.) Taking cash actually creates less of a paper trail which can lead to some less than positive happenings. When I first started in this business, the AD I first worked under would simply pay ticket takers and other on site personnel by grabbing some cash out of the ticket box and dolling it out to those people who had helped with the athletic contest that evening. Was that a quick and tax free way to pay your employees? Absolutely. Is it the right and legal way to do it? absolutely not. As a public entity we have to have or strive to have 100% transparency when it comes to how we handle the publics money in every way shape and form.
Just some thoughts from those of us on the front lines who actually do deal with this stuff