Theoretically, computer displays (and things like tablets and cellphones) use a separate standard usually referred to as sRGB, as opposed to 709. sRGB actually uses the same red, green and blue colour primaries as 709, so the colour performance is identical in terms of the deepest available shades. The luminance encoding is slightly different, so that if you grade something on a 709 display and then put it out on the web, the darker shadowy parts of the picture will seem a bit over-bright compared to how they looked on TV. The opposite situation quite often comes up when people grade on computer monitors then view it on a 709 display such as a TV, where the blacks suddenly look crushed. It's not a massively severe problem and people fail to notice it all the time, but you can fix it.
The problem occurs when people view YouTube material on a TV. Whether the YouTube servers notice what they're displaying on, or whether TV's YouTube app notices corrects the images is hard to determine. This is something that big studios hit all the time when they release trailers on YouTube and I've heard complaints about it.
As to LUTs. I presume your workflow will go somewhat like this:
- Record SLog3 in camera. Also consider which colour gamut you'll use. Often Sony cameras say something like "Slog3/SGamut3.Cine", and you need to know both.
- On set, monitor either using the camera's built in LUT capability, or using a display with LUT capability. You can use either a straight conversion to 709, assuming your display is 709, or something that's creative that you made in Resolve or whatever, but either way it's likely to output an image intended to be displayed on a 709 monitor. You can generally think of a creative LUT as converting whatever you're shooting to 709, via some creative decisions you made.
- In post, apply that same LUT in your grading app or in the monitor you're using while grading. If you did a really good job shooting it, the tweaks will be minor. If you had an off day, were rushed, or whatever, you may need to do more work, but the point is that you're actually working on the original untouched SLog3 material, so you have maximum flexibility.
There are a lot of side issues and complexities with all of this, so treat this as a starting point for discussion. What gear are you thinking of using, in terms of camera, monitoring and grading?