Alright. Here are some shots with three different lenses all at the same settings. To get this exposure I was at f/4 1/10 ISO800
The room is darker than you think. I've owned a few different "incident" light meters and I've tested the meters and several cameras. In all cases, camera metering and incident metering actually is _very_ close.
An incident meter is theoretically perfect because it isn't subjective. It's counting the photons landing on it's sensor. A "reflected" meter is based on light first bouncing off a subject and then into the metering sensor. But the reflection is skewed by the variable reflectivity of different objects. "white" things reflect more light than "black" things ... even if those things are in identical light. The camera is tuned to try to meter for "middle gray" (it's trying to correctly expose average things like skin tones, plant foliage, etc. Things which are neither "white" nor "black".
But knowing this... you want to be aware of what metering mode you use (evaluative, center-weighted, spot, etc.) and what you point the camera at to take the meter reading. If necessary, it is possible to point the meter at something that should give a good "middle gray" value, meter it and use the AE Lock button to "lock" the reading... then point at the subject you want to photograph to take the shot (and the camera will use the locked meter value. Just be aware that the "locked" meter reading is only locked until either (a) you take the next shot or (b) the camera time-out expires. You'll notice that when you tap a button to wake up everything... the camera stays in that state for many seconds, and things go dark again. Once things go dark, it releases the meter lock (it presumes you didn't really want to take that shot after all). If you want to force it to release the metering lock, you can tape the AF point selection button and that will drop the locked meter reading immediately.
Looking at your shots and your exposure... f/4 at 1/10th & ISO 800... and you were saying you had to use ISO 6000 to use f/5.6 and 1/250th. So lets do some math...
ISO 6000 is 2 2/3rds stops up from ISO 800 (800 -> 1600 -> 3200 -> 6400)
f/5.6 is down 3 stops from f/1.4 (f/1.4 -> f/2 -> f/2.8 -> f/4)
1/250th is about 4 2/3rds stops down from 1/10th (1/10 -> 1/20 -> 1/40 -> 1/80 -> 1/160 -> 1/320)
This tells me the exposure was different by about 1 stop (the shots you took at ISO 6000 f/5.6 & 1/250th) vs these shots you posted here at ISO 800, f/4 & 1/10th.
A slight difference in lighting, time-of-day (e.g. how much sun was shining in the window) and the metered subject in the room could all account for that.
The camera is pretty good at ISO 6000. You said the images were "unusable" (I shoot my 5D IV at ISO 6400 and don't even think twice about it. But you would want to make sure you understand how to use a workflow to process the images.
Here's an ISO 6400 shot from my 5D IV that I posted a few days ago.
A human eye is unlikely to be able to detect any noise in this image (all images technically have "noise" ... it's not a matter of "if", it's a matter of "how much" (specifically defined by a signal to noise ratio (SNR)).
You can find the original RAW files here: https://drive.google.com/open?id=1ZekVErNHUn7NU-RXSZWu5lnuQn--UkYv
(that's a Google Drive folder that contains both original RAW and JPEG versions of two images... one at ISO 100, the other at ISO 6400)
The trouble with the web posted version is that it's (a) resized and (b) converted to JPEG. Resampling an image to a smaller size means many pixels get averaged into fewer pixels and this tends to "average out" the noise. The JPEG conversion results in a bit more de-noising. I've noticed that when I compare the two JPEG images side by side, I cannot tell the difference. But when I compare the original RAW files both opened at zoomed up to 100% size (one screen pixel = one image pixel) then I *can* notice a tiny amount of noise in the shadows. THAT noise is EASILY handled in software (nothing I would ever think twice about).
ISO is only applied to an image AFTER the shutter is closed and the exposure is complete. ISO is not technically part of "exposure" ... even though you'll find lots of books & articles that talk about the "exposure triangle".
Anytime you have an insufficient exposure, you get noticeable noise. You can eliminate noise by getting a sufficient exposure. But remember that since ISO isn't really part of exposure... boosting ISO doesn't count toward getting "sufficient exposure".
There are ways to deal with noise in images and most photo-processing apps have tools for it (albiet fairly basic tools that tend to try to reduce noise globally). There are better tools that don't "globally" de-noise... they selectively de-noise based on tonality, use of image masks, or even let you "paint on" the de-noising (only de-noise the spots you select) and they produce a much nicer result.
If you post an example of an "unusuable" image as a RAW file somewhere (you wont be able to post it here ... you'd have to put it in a cloud drive somewhere and share the URL) I can show you what can be done with it. Also, what you *can* do is export the image as a JPEG with no compression and no re-sizing... but then "crop in" on some section you feel has too much noise (so that the cropped area isn't too big) and then post that.
Sometimes what people think of as "noise" is really something else.
