At 4 european universities i studied/taught this has never been the case. Most universities are used to run their infra, they ran their email servers way before google existed and they run big fleets of servers for thin clients. Afaik they still kept their own internal messaging as backup but it was still email servers hidden behind web gui.
What happened was that the big tech came in and made everything for them free. It is really hard to compete with free. They get windows for free, they get gmail for free at some point even unlimited google drive for free.
Now the situation is changing as the corps are tightening. I've seen 40k student university switch from gmail to office360 in two months because google suddenly wanted money and microsoft didn't. Now Microsoft also wants money. And it's not small money. So the school is doing cost assesment - you can give it to european third party provider that will be way cheaper tham microsoft. Or you go back to your own infra.
Turns out that what to be really expensive when google was giving people 30gb of free space to everyone in 2012 now is actually not that bad and you own your future. My guess is they will pay Microsoft for a year while they transition their email to their infra. The other parts gonna come later. But the students are required to use libre office (or latex) for writing their thesis so i don't think they see google docs as big blocker.
There's a huge difference between running an email server and some additional servers for thin clients -- all traditional stuff -- versus running an entire private cloud that redundantly stores the many many petabytes for your 40,000-person university, and all the web servers for the office software. Keeping it secure, keeping it updated, and having a live failover site if there's a fire or flood in your main data center that takes it out for weeks or months.
If it were that easy and cost-effective to do, large corporations would be doing it too. But there's a reason they're not.