Go. To. SKYWATCH! Its literally one of two public events that spawn there.

Wait! You trying to do Ikelos Fusion Core too? lol, I Literally waited on the moon for like 2 hours for the warsat event to spawn yesterday.

Some public events in locations are rare and don't happen 100% of the time. There are very few that have an almost 100% rate of occurrence (the Fallen Walker in The Divide is one such). Also note that Forgotten Shore is not the only place a warsat appears on Earth (and waiting on a public event there is rough because the warsat is 1 of 4 things that can occur then, meaning even if a public event happens, it's a smaller chance that it'll be the warsat). If you've not tried it, use http://destinypublicevents.com/ to try and help pinpoint places you should be. Please make sure you're there at least 2 minutes before it's supposed to hit (5 minutes is better) because the times fluctuate; stay there until 8 or 10 minutes after it's supposed to occur if it doesn't happen right on time. I've used it plenty in the past and should still be relatively accurate, just remember to get there early and stay late. Also take note that a warzone (where the enemies move against each other) IS a public event; it's just not forecast with the darkening skies, so if you see that occurring, that's the public event happening and it's not the warsat. If I remember right, your best bet for a warsat on earth is the Skywatch public event; it's either warsat or warzone.