Was Tohei in a match? If it's the one in that old black and white film I wouldn't call that a match. I'd call it using Aikido with someone who wants a match. Ueshiba invited the demonstration, hardly a match.
You could say everyone he ever invited to test him was a match but as he described pretty ceaselessly was about within yourself.......not to do with the other really. Hikitsuchi explains his words plainly enough I would say.
Inside you know if your intention is to beat, to win, to dominate etc. or if it's to give joy to, give life to, give comfort to etc. A high aim granted but those who do it successfully to any degree get an understanding of what no competition means.
My two pence.