What was the size of the company?
Generally speaking any organisation, given enough time, can establish processes that are better than JIRA with a certain amount of luck.
However, that's the best case scenario.
It's a standard distribution. Sometimes a 100 person project gets by fine with email. Sometimes it all goes to shit.
A tool like JIRA is, on average, a lot better than doing it via email only.
Small, 20 people maybe, but a lot of remote people that had their own small "domains" within the project. Sometimes one "domain" had multiple shifts (e.g. shipping). Communication between shifts was just done by writing yourself an email.
Granted, such a thing would not scale up to infinity, but for that size it was perfect and extremely hackable (just email + SMTP after all).
It worked flawlessly — in fact I haven't been at a place that was as well organized since. And it all came down to one thing: everybody understood that there is only email and that if it isn't an email it will be forgotten and fall in some crack, so you better make it an email, that has a propper subject, is searchable, etc. That meant everybody was writing emails with nearly military rigour, while the asynchronous nature of the electronic letter always reminded you, that you just needed your emails done and then you were fine. Depending on your position you had multiple (shared) inboxes that corresponded to the tasks. Of course there were some old emails, that were more of a "if anybody finds the time and will" type and sometimes emails from the guy from the day before had to be done etc. But all in all it was surprisingly pleasant to work just with email.
The point of the story is: You can turn any tool to shit if people use it in a bad way. And you can make anything work if people use it in the right way. Most problems with communications are people-problems, not tool-problems. Sure certain tools may lend themselves to certain behavior more than to others but ultimately a team can decide how they wanna communicate effectively and which behavior produces chaos and problems.