Or maybe not.
I've spent more time in East Shroud today than is probably healthy gathering the 15 legendary medals for the Yo-kai event. Very nearly all of that time was spent in the Sylphlands area. This gave me plenty of opportunity to keep tabs on What a Boar and to see if I could control the spawn since I was the only one clearing the FATEs.
I noticed something after running around for a little while. What a Boar, Everything's Better, and Breaking Dawn were never up at the same time. Seemed a little weird, so I stopped doing any FATEs but those three. Whether I let the current one expire or completed it, I never saw either of the other two until while the current one was active. Admittedly, I didn't let them run their course very often because I wanted my medals so I could leave. The time between one ending and the next one spawning was also variable. Sometimes it was immediate, sometimes it took a few minutes. Maybe even as long as five minutes (I wasn't very good with timing the interval).
I also never saw the same one spawn twice in a row. I can't say this is impossible, but it never happened for me. There was also no set order to which one spawned. Either of the two that wasn't just up had a seemingly equal chance to pop. I don't think I had to do Breaking Dawn and Everything's Better a combined total of more than 3 times in between What a Boar spawns.
I ended up with 9 tusks by the time I got my 15 medals and could have had a few more if someone else hadn't shown up and killed Kafre before I could get back there from doing one of the other two FATEs. You have a zero chance to make it from Breaking Dawn if someone is already waiting.
If this has been posted already and is common knowledge, I apologize. A quick search only returned posts lamenting how long it takes What a Boar to spawn whenever Boar Wars was drawn as a Sylph daily quest.
Happy hunting.