IMO, the relationship between Zen and Japanese Martial Arts, both classical and modern are tenuous at best. It seems to me is western Zen practitioners who see Zen in everything, even in Aikido regardeless of being based in a completely different spirituality.
Well the connection between Buddhism and the Martial arts tend to be though the more esoteric styles (Mikkyō) but as I understand it there is not that much difference in meditative practice so take that as you will. If one wishes to connect Zen meditation with their martial arts practice - and many have done so - no problem. Some of Ueshiba's major students did.
Ueshiba himself had some education in Shingon (Esoteric) but his main thrust was neo-shintoism which is even further removed - but we all ready know that.
What I found personally annoying is the insistence that closing your eyes in seiza is Zen Buddhism and that by adhering to this outer form somehow helps your aikido. I am all for a book which explains Zen in more depth and how it could be connected to your practice.