Not sure it undermines democracy. If by democracy you mean "get the counts right".
Paper ballots have to be counted by people. Lots of people. People are error-prone. And people could have agendas. Even if the risk that 1 person is making a mistake is 0.005% the risk is increased a if you have 5000 people counting votes. (It's not linear, but I can't remember enough of the statistics course to tell). This is the reason you want machines to do the counting. It's what computers do best. At least properly configured.
Using e-voting has nothing to do with "instant results", except that it's a bonus. It's to remove the uncertain, and boring, task of vote counting. I.e. people.
And is e-voting that expensive? Really? Compared to having thousands of workers and supervisors spend hours upon hours counting and recounting paper votes? I doubt that.
After the initial cost of the e-voting system, including bug fixing and so on, it's a "cheap" and re-usable system. Salaries of the error-prone workers probably outweight maintenance costs by a factor of ten. E-voting is a long term investment and staring at the initial costs is useless.
