I'm surprised that Dizzy Gillespie said that bebop was formed so that black men could play a style of jazz that white men could not play! Where did he say that?
Gillespie was a great player, with solid swing roots, and bebop in fact expanded upon many of the improvisational elements already prevalent in the swing era. It came about partly because of the decimation of the large swing bands due to many in their ranks being drafted after the USA joined WWII. Thus there was a move to smaller swing units, and from this the seeds of bebop were sown (at such venues as Minton's Playhouse and Monroe's Uptown House).
In fact, I think that the roots of bebop can be traced to way before the “official” start of the movement, and I am inclined to agree with those who feel that Coleman Hawkins’ “Body and Soul” was one of the key precursors of the bop movement. Actually, in the same year (1939), Gillespie recorded a seminal solo on Lionel Hampton’s excellent “Hot Mallets”, which many believe to be the first recorded example of what would later be termed bebop (Hawkins was on the session too). This illustrates the fact that nothing is chronologically clear cut. New genres develop over time, they don’t just start! And they are certainly not invented.
The complex harmonic sense required to perform bebop did leave many swing musicians (white and, indeed, black) behind, but there were new generations of musicians to take up the musical challenge, including white musicians. Gerry Mulligan and Gil Evans, for instance, were almost in at the start of the bebop movement, and Mulligan worked with Miles Davis in a highly productive manner. Gillespie also worked with many white jazz men, and was even briefly with the Stan Kenton big band (Gillespie never lost his big band roots). This was cross-cultural empathy resulting in artistic achievements of the highest order. You’re certainly right that jazz wouldn't have flowered into so many forms that we know today without this cross-cultural cooperation.