I'm not a pro so just going to share my experience when I sold my last home. The inspection uncovered mold in the attic (to my dismay.) The mold remediation company + builder friend + HVAC friend all agreed it was due to not enough venting in the attic. Turns out that when insulation was blown in, some of it covered the baffles coming from the eave vents (my vents looked like your second picture, but my eaves were quite shallow.) So, in addition to remediating the mold, I had a new vent fan installed, higher baffles put in, and the insulation raked to ensure it doesn't cover the baffles. I was super stressed when the mold was discovered by the inspector, but the fix wasn't as difficult or costly as I had feared. (I just wish I had figured it out before I put my house on the market!)

So my very amateur advice is that yes, trapping hot air could cause a problem. I'd play it safe and have venting plus a fan. Note that I needed to have an electrician run the wiring to the fan, since the roofers who installed the fan didn't do that part. YMMV.

So it turns out I got some of the info wrong in my original post. It turns out that I have vents in the eaves that were covered up by the sofft panels. The installer says the soffit panels aren't air tight so there is still some air flow into the attic, but in my mind the closed eaves will significantly reduce air flow to the attic.

He has offered to install 2" round vents into the soffit panels to increase air flow. I don't know if this is adequate or how many of these little vents I'd need to create the right amount of airflow.

Any thoughts on this situation and what I should do?

Typically every rafter bay is required to have a vent today as well as a ridge vent or roof (mushroom type) vents. One by itself does not vent.

@millworkman Every rafter bay? If I'm understanding that right, that sounds like A LOT of vents. Anyway, this house isn't remotely close to new construction so if that is a new standard I can't meet it, I just need to do whatever is possible to make the best of my situation and avoid future problems.

I have seen many references to the 1/150 rule for attic venting. That is Section R806.2 of the IRC. Isn't that the standard? I estimate my second floor attic is 1,100 square feet which translates into about 7 square feet of ventilation, and the vent in the photo above probably covers about 3 feet. That leaves about 4 square feet of vents that were covered by the new sofft panels and are no longer free flowing.

So given that the soffit panels aren't air tight, it seems like I have to figure out how much venting to add to the soffits to allow enough air to flow through them and into the existing attic vents. Am I off base?

There's a difference between building code requirements and best practices. The code development cycle is long; typically a building code is adopted three years after its publication. Published code requirements typically lag best practices by a number of years.

With regard to attic ventilation, you'll want approximately 50% of the vent area to be met by soffit vents and 50% to be ridge or similar vents. Gable end vents--like the ones in the photos--aren't all that effective for ventilating attics.