Vehicle: 2010 Ford F250 with a 5.4L Triton V8 Miles: ~45,000

I have a ticking 5.4L engine in an F250. This started a few months ago and I've been trying to figure out what is wrong with it to no avail so far. It seems to be coming from the driver's side of the engine bay.

The sound is most noticeable immediately on startup and then quiets down once the engine revs down. It is also noticeable when driving, specifically when accelerating above 1,500 RPM. Letting off of the accelerator makes it stop from the perspective of sitting inside the truck. However, it is also not noticeable in neutral when revving the engine above 1,500 RPM. So it does seem like it's a function of RPM, but only when under load and on initial startup.

It's hard to record the subtlety of a noise like this, but here's a video of it during startup, then going down to idle, and finally me revving the engine a few times: https://drive.google.com/file/d/15EsTdjziC7ycqt_eUJEHhpUpe3aDoAha/view?usp=sharing

All of this leads me to believe it's either an exhaust leak or a timing/valve issue (since this engine is notorious for those). Here's what I've tried so far:

• Installed all new roller followers and lash adjusters (made no difference)
• Installed new spark plugs (made no difference)
• Inspected the exhaust for leaks with a leaf blower connected to the tailpipe (could not find any, including on the exhaust manifold)
• Started engine with the VCT solenoids disconnected (made no difference)

Aside from doing a full timing job, which given the truck only has 45,000 miles on it doesn't seem like that would be necessary, the only other thing I've noticed is that a stud has broken off of the exhaust manifold on the same side the noise seems to be coming from. Photo: https://i.imgur.com/y4HLEAb.jpeg

So it seems like I have two options left:

1. Replace the exhaust manifold. This seems like a real pain in the butt to get off and I'm nervous to drill into the engine block to get the broken stud out so I've been avoiding this.
2. Continue on with a full timing job including new phasers, tensioners, and chain. It's not clear to me how this would solve the problem though.

Any advice?