I have run a distro which doesn't have syslog nor cron installed by default. I felt lost at first, but when I learned how to read and work with the journal and systemd timers I found I didn't really need syslog and even cron. Ubuntu uses both the syslog and the journal and some timers and a few cron jobs. How would I go about removing system logging and even cron on Ubuntu without breaking something? I'm not asking out of necessity at this point, but rather to get rid of processes I do not not need.
Ubuntu has tens of thousands of files you don't need but are not worth trying to get rid of.
Ubuntu is really an ecosystem of packages. Some of those packages have also switched to systemd, but others may use Upstart, SysV init scripts or cron jobs. You could turn off cron jobs and it might not break anything right away, but you later might end up installing something which expects cron jobs to be running. That said, you can disable cron jobs like this as long as you willing to accept that some things that expect to run might not be run:
sudo update-rc.d foobar disable
Likewise, some packages may expect to log to syslog instead of the journal. You can remove
sudo apt-get remove rsyslog
But I wouldn't recommend doing that either. I think the adage applies: "Benchmark first, then optimize." First find out what's actually slowing down your system in a meaningful way and work on fixing that. I doubt that cron or rsyslog will be your biggest problems.