Directly Contacting Hiring Manager / Team Leader

I do some hiring for my team and it comes across as annoying, but I'm also the kind of guy that doesn't like follow up emails thanking me for my time. That said, I did give a kid an interview that probably wouldn't have had one otherwise (with a higher than normal bar because of my annoyance) because he basically messaged me, my coworker, our CFO, and posted publicly on the job position that he was interested.

I would just send in a friend request and let them know you applied

I don't think there's any real harm in doing what you described. Others may disagree, but I say go for it.

Now what I find weird/awkward: I recently applied for a role on LI, the hiring manager (CEO of the firm) immediately added me as a connection upon seeing my application & no further outreach on his end after that. So this dude saw my resume, decided to add me as a connection, but didn't feel the need to give me a phone screen...

The key to doing this elegantly is with a warm intro from a mutual connection if at all possible.  Most folks probably won't dismiss you out of hand for this and will likely just forward your shit to HR.  There is a small risk of blowback if you come across as annoying or too desperate.  

Currently going through the process, I usually look at the company and see if there is anyone with a connection and try to use that.

If not, I'll go to similar position (i.e. analyst/associate on the team and it's an analyst position) and reach out saying I'm interested in learning more about day to day or some bs. I've also done that on Linkedin asking for a call. Just introduce yourself, where you went to school, what you've been doing and do they have time for a call. You can do this on Linked with a private message (get Premium) & add them or find their email format.

I've only seen one say not to contact, then I wont, but otherwise I will because worst case you're networking/meeting new people for the future and best case you get an interview.