Hjones....I think you are discussing a step improvement when you should be considering a path. If you are happy with the 60D, then you should look to supplement that kit and build around a crop sensor body. If you think you will go FF someday, you likely want to start putting together the pieces and having that path.

So, to try to give input to your latest questions directly. No, given that you describe using the 55-250II for outdoor sporting events, I do not think it would be a good idea to limit yourself to the 6D and the 24-105. I think you would find yourself missing the >105 range. Also, you mention portraits, and the 24-105 is nice, but something with a wider aperture would be better.

So, based on what you describe, I would be most tempted to upgrade from the 55-250 II to the 55-250 STM (which is getting very good reviews). For the very generous offer from your friend, I would consider a prime lens such as the Canon 35 L/35 f/2.0 IS, Sigma 35 f/1.4, Canon 50 L/50 f/1.4, EFS 60 f/2.8 macro, 85 L (probably too much)/85 f/1.8. The intention of these would be to help with your low light/indoor photography. I would look at the focal lengths that you tend to use the most and pick a prime that matches that length. If possible, adding the 35 f/2 IS or Sigma 35 f/1.4, 50 f/1.4 and 85 f/1.8 would give you a very nice prime set up, some great optics and still be under your friend's offer of $2k. I own the 50 f/1.4 and have evaluated the other two at different points. The 50 f/1.4 is ok at f/1.4, sharp by f/2, and amazing by f/2.8. Even though I have several "L" lenses, I've been noticing that a lot of my favorite pictures recently have been coming from my primes (50 f/1.4 and 100 mm L). Both are very sharp, but it is also the low light that I can work in with those lenses.

Regarding the path, on a crop body, the 35/50/85 would cover your "classic" portrait range with fast glass. If you moved to FF, they are still very usable. The issue with primes is flexibility. So this makes sense if you are willing to carry a couple of lenses and switch them out. So if you are a more methodical shooter. If you just want to walk around with one lens on your camera body, then I think you are back to the 17-55 f/2.8 or possibly the 24-70 II.

Good luck.