In my experience, the SmallRig Cage for Sony Alpha 6700 4336 is the way to go cuz it doesn't block the flip screen. Heres the best setup:
* Grab the SmallRig NATO Top Handle with ARRI 3/8"-16 Locating Holes 2439B for fast assembly.
* Use the NATO rail for easy breakdown. Honestly, it's the best value for run-and-gun work... you wont regret it!

So, been thinking about your setup... I've been shooting with these Sony APS-C bodies for literally years, and honestly, the a6700 is a beast but it's sooo small. Basically, when you start rigging it out for video, you realize that space is at a premium. I've had issues with some generic cages in the past that promised full access but ended up making the dials almost impossible to turn with my fingers... super frustrating. You gotta look at the "skeleton" of the build first. A cage isn't just for mounting stuff; it helps with heat dissipation and adds structural integrity. If the tolerances are even slightly off, the camera wiggles inside the cage, and that ruins your footage highkey. It's all about how the cage distributes the weight of your monitor and mic without stressing the camera body. For your situation, I'd suggest the Tilta Full Camera Cage for Sony a6700 - Black TA-T54-FCC-B. It's actually really sleek. Unlike some other brands, Tilta's design leaves the right-hand grip mostly exposed so you dont lose that ergonomic feel, which I love. Plus, it has an Arca-Swiss plate built into the bottom which is a HUGE time saver for run-and-gun stuff. For the handle, definitely grab the Tilta Compact NATO Top Handle - Black TA-QRTH7-B. It's got that NATO rail for quick teardown. I find Tilta's NATO clamps a bit more secure than the generic ones I've tried. It's got the ARRI locating holes for your monitor mount too. Unfortunately, some cheaper handles I've used lacked the proper spacing for the pins, which was not as good as expected when things started spinning mid-shot... total nightmare. This combo keeps things LIGHTWEIGHT tho. gl!

Honestly, I am in the exact same boat right now and it is driving me crazy. I have been looking for a solid setup for my Sony a6700 for like three weeks and I still havent pulled the trigger because every cage seems to have some weird compromise. Like, one brand looks great but people say the screen clearance is tight, and then the other brand looks better for the screen but looks way too bulky for a compact camera... I just want something lightweight that doesnt get in the way of the dials or the battery door but it feels like the perfect balance doesnt exist yet.

Tbh, looking at the market research on these rigging ecosystems, you gotta be careful with the NATO rail tolerances. I mean, not all rails are created equal—some brands have slight variations in the dovetail width that lead to serious lateral play with high-end handles, which basically ruins your stabilization for run-and-gun. Another thing people overlook is the clearance for the Multi-Interface (MI) shoe. Since the a6700 relies on that for digital audio, some cages have an overhang that blocks larger Sony mics or XLR adapters. Its a huge oversight in the CAD design of some of these units, right? Also, watch the micro-HDMI port. If the cage doesnt include a dedicated, adjustable cable strain relief, youre just begging for a board failure during a shoot. I've seen too many form-fitting cages that actually restrict the flip-screens rotation once you have a full-sized cable plugged in. Basically, the technical trade-off usually comes down to whether they prioritized the aesthetics or the actual I/O clearance. Just double-check the hinge clearance specs before you pull the trigger, you know?

Seconding the recommendation above! Tbh, sticking with the big name brands is SO much better for fitment than cheap knockoffs I've tried over the years. In my experience, budget cages often block dials, whereas name brands fit perfect. You could also look at half-cages since they're the lightest and cheapest way to rig up. Are you planning to mount a heavy monitor or just a mic? Knowing your budget would help too lol.