Stressed joints are much more likely to slip over time, or in this case to separate further. Considering the purpose of this glue up I would, at a bare minimum, cut a groove down past the part where it is separating and put a strip of wood in.
Some of the best advantages of hand tools are that you can fine tune a fit up with a hand plane that would be otherwise difficult with machine tools. Even if I am cheating by using a lunchbox planer to get the stock to rough size I will work the remainder of it with hand tools to ensure the surface is flat. You can, of course, do this with a jointer (machine) but it's easier to work the fine end of the spectrum with hand tools.
To to trim the end grain I often do not pare, Instead I will clamp a block, which is perfectly 90 degrees to the area in question and chop it with a mallet while holding the chisel tight to the guide block. A dull chisel will make a mess of this, so it's important to use a sharp one.
Mark with a knife prior to doing this, so that you can use that to start your cut.