I've got a new camera and a lot of stress, so I barely had time to set it up completely before the shoot (nothing professional though anyway).

Tracking moving subjects is a skill that takes practice just to become minimally competent. Panning at slower shutter speeds to keep your moving subject (mostly) blur free while the background is blurred to show the motion of your subject is even more difficult. Not only is knowledge of how to do it required, but muscle memory needs to be developed. It's more like actually shooting a basketball than like diagramming a basketball play.

I've known some amazing photographers in terms of composition, lighting, exposure, etc. that can't take a decent slow-panning shot to save their life. Most of them only try slow-panning shots once in a blue moon while on assignment, get frustrated very quickly with the unsatisfactory results, and go back to using a fast enough shutter speed to freeze all motion.

Which camera mode do I use for moving subjects?

Which exposure mode one selects should be based more on the lighting conditions and how stable or how rapidly changing they are than on whether one's subjects are moving or not. The desired end result may also play a role.

Which autofocus mode one uses should be based on the amount of movement by one's subjects. 'AI Servo AF' or 'AF-C' (depending on what brand the camera is) is generally the best AF mode to use with moving subjects, but there are some rare times where the nuances of the situation might make 'One Shot/AF-S' a better choice.

Which AF area do I use for moving subjects?

This highly depends on the specific camera, what AF point selection methods that camera offers, and how well each choice performs in that camera.

For your EOS 6D Mark II, if the "Zone AF" option that gives equal weight for a square of nine AF points¹ works for you, I'd stay with it. I'd expect it may be more problematic if there are other things in the foreground closer to the camera than what you wish to focus on.

Using Large Zone AF or Auto selection AF, I feel like, with a lot of possible subjects in the frame, my camera would just not do the right job of picking who I want to pick, so those were out.

I'd recommend you try using 'Automatic selection AF' and using [C.Fn II-11: Initial AF Pt in AI Servo] set to either option 1 or 2 that allows you to designate the initial AF point.¹ If your subject is at the selected AF point when you initiate AF, the camera will attempt to track that subject as it moves around the entire frame.

Disclaimer - I've not tried using 'Automatic selection AF' that allows the user to select an initial AF point with a camera, such as your 6D Mark II, that does not also have an RGB+IR color meter and EOS iTR AF.² YMMV.

For your EOS 70D, I'd probably avoid 'Zone AF'. 'Zone AF' is notoriously problematic for selecting the closest thing to the camera, even when that thing is outside the selected zone with the EOS 7D. The more recent EOS 70D uses the same PDAF sensor as the 7D did and most likely uses many of the same AF algorithms. Instead, I'd go with '19 point automatic selection AF' in 'AI Servo AF' and use the method outlined on page 105 of the Canon EOS 70D User Manual to select the initial AF point.

¹ Please see pages 139-40 of the Canon EOS 6D Mark II User Manual

² I've personally found that 'auto' selection works well with the upper tier models (1DX, 7D Mark II, 5Ds, 5Ds R, 5D Mark IV, etc.) that let one select the initial AF point manually when using 'Auto' AF point selection mode with 'AI Servo AF' and also offer EOS iTR AF(EOS Intelligent Tracking and Recognition Autofocus) which uses a dedicated processor to combine distance information from the AF sensor with color information from the RGB+IR metering sensor to track moving subjects when EOS iTR AF is enabled in 'Auto' AF point selection mode.