Why some emails dragged to a tab not showing in Gmail?

My Gmail is provided by my college’s G-Suite. I have just subscribed to a newsletter using this email account. I dragged these emails to a Socials tab, then made the Socials tab appear on the Gmail homepage. But when I click on the Socials tab, it doesn’t show these emails. It shows the other Google Classroom emails on Socials though. I have to click on Socials on the sidebar to make these newsletter emails appear.

I tried dragging them to Promotions too, still doesn’t work. But when I click on that tab from the sidebar it shows? Why does this happen? Is there a workaround? Is there any other way to add them to different tabs besides dragging? Something that’ll make it work?

I tried the option of “Move To” to Promotions, but still doesn’t show up. When I do “Move To Inbox”, only then they show up. What am I missing here?

What is this Inbox? There’s no tab for Inbox.

Also, how to automatically have all emails from this sender be sent to my Promotions tab? And show up too?