What is the best way to join toe kick boards?
Don't join Them
Well the best option is to only have one kickboard.   Looking at yours I would just throw them away.   For kickboards I use 1x and I use either pine or Azek.   That is the area of the kitchen that has the most chance for water damage so don't use plywood or something like that.   MDF would be your worst choice (isn't it the worst choice for almost anything).
If you do... do it with style
The other option and just used this for the toekick on a rather big island is a rosette.  If you center this it becomes a design element not a stop-gap solution.   I used something very similar to this - link.   Note that rosettes are often thick... So you may have to plane the back.   I usually put the toekicks next to the rosette.   You will not be able to do that if the boards you have are as thin as you described.
Also my suggestions are based on the other materials being used...  Your cabinets look really nice.   Throw those toe kicks away and go with a long piece of pine or azek (any pvc 1x6).  If you need to rip the 1x6 put the ripped edge on the top - no one will ever see it.