Tex wanted the fans to see a variety of colors. That's what I recall.
Tex was the ultimate marketing man, and the marketing behind the white jerseys was brilliant. Back then, the NFL wasn't the monster it is today, and if you built an NFL franchise, there was no guarantee an audience would show up. You had to actually sell game tickets to make money.
The reason Tex wanted visiting teams to wear their home jerseys was that he recognized, in the early 1960s, the Cowboys were an unknown commodity, and his best audience for buying tickets would be people who grew up in other NFL cities. This audience was used to seeing their teams play in their home jerseys, which were usually the colored jerseys. So he created an environment in which sports consumers could buy a ticket to the Cotton Bowl to see the established NFL teams in their home jerseys as they played Dallas's new franchise.
The team quickly realized that, playing in white jerseys offered the extra value of being a little cooler in the Dallas heat. But now that we're playing under a roof, that's (sadly, for a traditionalist) no longer a factor.
Texas E. Schramm, isn't just a hero of mine as a Cowboys fan, but he's an idol as someone who has spent 25 years working in advertising agencies. He not only played a huge role in the Cowboys organization, but in the way football is played on TV, in the creation of Monday Night Football and in marketing any sports league to consumers. If you're ever in a bookstore and see a Tex biography (there are several good ones) on the discount shelf, I definitely recommend buying it and reading it several times.