I can only speak of the gutter mounted.
EAG is becoming a bit more popular an option for three reasons:
You don't have to drill the roof
You can leave the rack on when you remove the roof
There are many naysayers on this site that say no to gutters. Primary reason is "the gutters will fracture" and the factory did not rate the roof for a rack on the gutters. The factory did not rate the front suspension for 4" lifts and 38" tires but people do that on the daily without upgrading every component on it.
I mounted an EAG JK (smaller) rack on my JKU (longer) color matched top with about 7-8,000 miles on my rig. I am now approaching 50k and there is nothing wrong with the roof or rack. I am taking it off this week to reweld a spot on a flat plate I welded on the back for fuel and water storage that is vibrating. Shitty weld placement on my part. I will take pics of the mounting spots and post them to show that there are no issues.
I have taken mine to ALBQ/SF twice, Ouray/Silverton twice, Flagstaff, spent 2 months in Yellowstone/Montana, recently to Terlingua and Cloudcroft, etc. with ~160lbs+ of weight on the rack. Fuel, water, 3 Plano bins with camping gear, food, traction boards, ax, shovel. Not a single issue that I am able to see.
As for leaving it on while you remove the top, I would not necessarily lift by the rack, but if I did, I would not leave it suspended.
Great rack, with the JK on a JKU version you can still remove the freedom panels without issue, ships from LA but clearly made overseas, Mounts easy, 1 person install in about an hour or so.