A couple of points to the very fine comments already. Your choice really depends on how you prioritize your use for this lens.
A previous poster explained that IS is less important than fast/good autofocus; when it comes to photographing children, I totally agree. If this is important to you then the 70-300L becomes a stronger choice because it has newer, more advanced AF electronics that work better with your 5D3. Roger at Lensrentals has documented this in abundance; look for his blog entries on autofocus. With the 70-300L and your 5D3, you can shoot at higher ISO/higher shutter speed and have better autofocus response. It is also a very compact lens that is very versatile for landscapes, etc. While some may quibble with its optical performance compared to other canon L lenses, I find it to be excellent and functionally equivalent. I own a 135 F2.0L and 200 F2.8L and find the images from the 70-300L are equally usable in terms of sharpness and overall image quality.
If portraits are really your thing, then consider a 135 f2.0L the extra lens speed and handling will be a better fit for portrait situations. If you are looking for longer reach, then consider a 200 F2.8L - this lens is superb.
Probably the best advice you received is to rent one or more of these and try it for yourself. Lensrentals makes it easy with good selection and fair prices (no I don't work for them but have rented from them)...The tradeoffs in the answer to your question are subtle enough that all of our recommendations on this forum will only take you so far...you have to experience them for yourself.
Best of luck in your decision.