The Fuji lenses I was calling big and heavy are their medium format line. They're using small medium format as a response to 40+ MP full-frame, and it's an interesting choice, but held back by big lenses. The 5 year old 50 MP medium format sensor is pretty much directly comparable to the newest 46 MP BSI sensor found in the D850 and Z7, and has only a small advantage over the 42+ MP sensor in the A7rII and A7rIII (from comparisons online and from looking at large prints from all three at PhotoPlus - the only one of the three I have direct experience with is the Nikon, which I chose after careful comparisons of output). Only the Nikon version of the 40+ MP Sony BSI sensor has ISO 64, which brings DR and noise that last little step to compete with the medium format sensor. When Fuji gets ahold of a newer MF sensor, their choice may very well pay off.
The Nikkor Z 24-70 is a very sharp lens except in the extreme corners (so far out that they're often hidden under the mat board, and can be cropped out very darned easily). The corners aren't horrible - this is not an APS-C kit lens... Think 24-70 f2.8 or better (your choice, Nikkor, Canon or Sony G-Master) across most of the frame, with corners from a 24-120 Nikkor or some other decent compact FF lens. Don't worry about the corners - marvel at 4x5" image quality in a small package with weather sealing and stabilization. The Z7/24-70 is the sharpest camera/ lens combination I've ever used (medium format and A7rII or III are outside my experience, although I've used an original A7r quite a bit), and by a significant margin, too.
I don't consider Fuji APS-C lenses big or heavy at all - in fact, Micro 4/3 pro lenses are excellent, but big and heavy by comparison to Fuji lenses, especially given that the sensor size gives up a stop. The best example is the Olympus Pro 12-40 f2.8 vs. the equally good Fujinon 18-55 f2.8-4. They're both excellent standard zooms, but the Olympus is a bit bulkier and heavier (about 25%) despite the smaller format. Olympus fans will say "that's not fair - the Olympus Pro needs to go up against the bigger and heavier Fujinon 16-55 f2.8, not the 18-55 f2.8-4".
Once you consider the sensor size, the smaller Fujinon is actually a faster lens in most respects (by a full stop at wide angle, part of a stop through much of the zoom range, and they're the same speed at full extension). The Fujinon 16-55 f2.8 that is sometimes cited as the appropriate comparison is a full stop faster throughout the zoom range. This applies both from the viewpoint of bokeh and subject isolation and from a noise and image quality standpoint.
Is the Olympus Pro a "better lens" than the Fujinon? The proof of that would be in which lens one could make an image that looked better in a print with - whether it printed larger, or was more to your taste at the same size... I have never used a Micro 4/3 camera with the most modern 20 MP sensor seriously, but I've used that Pro lens on the 16 MP sensor extensively. It's a very good lens, and I've gotten good results out of it, but I've gotten better results out of even the 16 MP Fujis (X-T1) with the 18-55. The latest 24 MP Fujis (X-T2 and X-H1) are in a different league - they offer substantially more detail and less noise with the 18-55 mounted on them. Comparing the lens and sensor combination may not be fair, but it's reality - a Micro 4/3 lens could be perfect, but it will go on a sensor that gives up quite a bit to larger sensors and it cannot go on another sensor.
That doesn't mean that Micro 4/3 always loses, either. If you had a Nikon D3500 with a very similar sensor to the Fuji, but with the Nikkor 18-55 kit lens, I have very little doubt that the Olympus Pro would win easily. Even more so against Sony APS-C with their usual kit lens - that 16-50 pancake is terrible (and DxOMark, etc. point that out - DxO says it resolves 6 "perceptual megapixels", the lowest score of any lens). I've never really shot Sony other than a few borrowed cameras (some for significant periods, though - I have a friend with about four systems at any given time, and I've grabbed his Sonys for many hundreds of images, including taking one home for weeks) , but I was shocked when a student once handed me his a5100 with the 16-50, and I could see the lens distortion in the viewing screen before taking the picture - that's hard to do, other than with ultra-wide lenses (fisheyes are, of course, the extreme example), deliberately distorted lenses, or by staring at graph paper.
Yes, there are better Sony APS-C lenses than that one (all the rest, one hopes), but that's what you tend to get with a body, even the a6500. They have discontinued their 18-55 (just like Canon and Nikon's cheap 18-55s) in favor of the even worse 16-50. I just checked Sony's site 5 minutes ago, and they offer the a5100 and a6000 only with the 16-50 or as body-only. With the a6300 and a6500, the 16-50 is the first option, but you can also get an 18-135. B+H will sell those bodies with a much better 16-70 Zeiss, but there's no discount compared to buying body and lens separately. Sony's pushing the execrable 16-50, so it's fair to compare...
I've used the Sony 10-18 a few times - my friend with various cameras has one, and it's a very nice lens. I prefer the Fujinon 10-24 for the range (the overlap with longer lenses between 18 and 24 is nice), but they are certainly comparable in image quality. I've never seen a 16-70 Zeiss except on a shelf, but it could very well be great (it should be, for the price - it's comparable in price to the Fujinon 16-55 f2.8 when there's a good rebate on the Fujinon, so that's the standard it needs to meet) - if it is as good as that, Sony has something that Canon and Nikon don't - a really good dedicated APS-C standard zoom. Wish they'd put it in the kit with the higher end bodies and give a nice discount on it.
Micro 4/3 does always lose in overall image quality at base ISO against a larger format with a comparable lens. There are numerous caveats in that statement, though!!! I'm far from a Micro 4/3 basher - they don't fit my shooting style or subject matter, but I think they're among the most interesting cameras out there. The most general caveat is with a comparable lens. Below full-frame, only Fuji has as good a lens lineup as Micro 4/3 - I'd far rather have Micro 4/3 with a good lens than Canon, Nikon or Sony APS-C with a kit lens (any comparison of kit lens to kit lens without either a Fuji or a full-frame camera in it can effectively ignore the sensor - none of the cheapies come close to what the sensors can do).
This is made more interesting because Canon, Nikon and Sony all have APS-C lineups made up largely of kit-type lenses (several variants of literal kit 18-55s plus cheap tele zooms and various superzooms). All have a few better lenses that break that mold, but none have anywhere close to a full line of them (Thom Hogan loves to criticize Nikon for this, but Sony's no better and Canon's only a bit better). Fuji has the only full line of decent APS-C lenses...
Another place where Micro 4/3 often wins is when nothing else will get the shot (or only a few particular "something elses", all of which are much heavier and/or more expensive). The E-M1 mk II in particular is an unusually fast camera with a really special image stabilization system. You can't get that night cityscape at 1/2 second with anything else other than a tripod... The action shot at a sporting event? An E-M1 mk II has a better chance than anything except a D5 or a 1Dx mk II...
The third advantage of Micro 4/3 is at the extreme edge of compactness. Different bodies (the best Micro 4/3 bodies are the size of APS-C or even full-frame mirrorless, but there are ultracompact bodies that give up weather seals, battery life and even the viewfinder), different lenses (no lovely Pro lenses need apply), but you can build a compromised kit the size of many compact cameras. Yes it's compromised compared to most interchangeable lens cameras, but it's better than any compact camera (except a few exotics - Sony Rx1 series or some Leicas).
