The weight difference between Micro 4/3 and Fuji APS-C or even Sony, Nikon or Canon full-frame is tiny when you look at the bodies and lenses you'd want. Yes, you can come up with some theoretical Micro 4/3 systems that are tiny and really light, but you lose a lot of the advantages. The really small bodies are NOT the incredibly durable ones - the E-M1 mkII is as durable as any camera out there, but the E-M10 line and the PEN line (where the real weight savings are) are fragile, non weathersealed consumer cameras. The incredibly durable E-M1 mkII is 75 grams heavier than an X-T2 and only 100 grams lighter than a 46 MP Nikon Z7.
The same holds for the lenses - the Olympus pro 12-40 is heavier than the (comparable quality) Fuji 18-55 - yes, the Olympus pro is a fixed aperture, but from a depth of field perspective, the Fuji is actually faster at the short end. You could replace the Olympus pro lens with a very high-quality prime (you could come up with a similar weight savings with a Fujinon prime, although the Micro 4/3 lens would be lighter), or with a low-quality consumer zoom, where the Fuji 16-50 isn't that much lighter than the high-quality 18-55 but the Olympus options go down to a pancake lens that weighs little more than a body cap (but most of the reviews I've seen say image quality is also close to what you'd get from a body cap).
Here are the weights for a bunch of options as close as I could get them for a quality, weathersealed body with a high-quality standard zoom and a wide zoom. The comparisons are NOT perfect - nobody makes perfectly aligned gear. Sony and Canon have a little extra reach on the telephoto end, because the only Canon lens in the same focal length range is a very heavy f2.0, and the Sony option that doesn't go to 105 is the 24-70 f4 "Zeiss", which doesn't perform as well as the others (it's in the notes). The Panasonic GH5 and G9 are both heavier than the Olympus E-M1 mkII. The E-M5 mkII is in the notes - I chose the E-M1 mkII as the primary body because its build quality matches or exceeds the others.
The Micro 4/3 lenses being f2.8 is not an actual discrepancy - it's needed because they have a full stop less subject isolation than an APS-C lens of the same speed and close to 2 stops compared to full-frame, and the sensor is at least a stop noisier/less DR at the same ISO. A Micro 4/3 body shooting a f2.8 lens wide open at ISO 100 is a very reasonable comparison to a Fuji at f4/ISO 200 or a modern full-frame sensor at f5.6/ISO 400. There are no high-quality Micro 4/3 f4 zooms, either - once you leave the excellent f2.8 lenses, you are looking at cheap consumer glass that is f5.6 or worse at the long end.
The batteries aren't quite comparable - the Fujis and the E-M5 mkII use a smaller, lighter battery than the others, the E-M1 mk II is in the middle, while the Sony, Nikon and Canon use beefy DSLR batteries that get around twice the shots per charge. Add 50 grams or so to the Fuji and the E-M5 mk II for a spare battery to compensate.
Fuji X-T2 (503 grams), 18-55 (308 grams), 10-24 (404 grams) - 1215 grams with 1 battery and card (add 173 grams for the X-H1, which has IBIS like the others).
Olympus E-M1 mk II (574 grams), 12-40 Pro (382 grams), 7-14 Pro (534 grams) - 1490 grams with 1 battery and card (save 170 grams by substituting E-M5 mk II, but go from probably the most durable body in this group to probably the least).
The lightest Micro 4/3 kit with a similar focal length range (the wide angle doesn't go as wide) is much lighter - 650 grams, but uses severely compromised lenses. E-M5 mk II body (404 grams), 14-42 f3.5-5.6 body cap lens (91 grams), 9-18mm f4-5.6 (155 grams). The 14-42 is a very weak lens, and the only superior options are the relatively heavy Olympus 12-40 and Panasonic 12-35 f2.8 lenses.
Nikon Z7 (675 grams), 24-70 (500 grams), 14-30 (~500 grams) - 1675 grams with 1 battery and card. The Nikkor 14-30 exists only in prototype form, with no released weight. It is almost identical in size to the 24-70, so I used the same weight.
Sony A7r mkIII (657 grams), 24-105 (663 grams), 16-35 f4 (~518 grams) - 1838 grams with 1 battery and card (the "Zeiss" 24-70mm f4 is approximately 250 grams lighter, but a widely recognized much weaker lens).
Canon EOS-R (660 grams), 24-105 (700 grams), hypothetical 16-35 (541 grams) - 1901 grams with 1 battery and card. The "Canon RF 16-35" is a nonexistent lens - I calculated what it should weigh based off the Canon 16-35 f4L DSLR lens. The 24-105 RF is lighter than the 24-105 EF, so I subtracted the same percentage (almost 12%) from the weight of the 16-35 EF to get a potential weight for the RF wideangle zoom.
The weight difference from the lightest (Fuji) to the heaviest (Canon) option is about 35%. Once you add the weight of the extra battery to the Fuji, it's around 30%, and if you use the X-H1 body to gain IBIS, it's only about 20%. Micro 4/3 is in the middle of the range, although it has the most room to substitute lighter components at a cost in image quality and durability.
