I think the only real problem with such an idea is nothing to do with using Discord or Skype et al. but the timezones some of us are in.
Other than, from my recent experience with it, Discord works out pretty well for running game sessions.
A mate of mine is running a Call of Cthulhu game via Discord but we're all in the same city so timezone doesn't matter. However if I was to try and jump on to something being run in a North American or European timezone, it most often puts my local time as something ludicrously early in the morning (like, still dark outside kind of early in the morning!)
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