Now we're just splitting hairs over the nature of the support, whether it was intrinsic or it came through cajoling (the latter of which has always been part of lawmaking). And it was a Democrat president for whom civil rights was so important to do all the cajoling to get it pass.
Curiously, the bedrock of opposition to civil rights in the 1960's came from the same people who oppose LGBT* rights today (and the most loyal Trumpers) — Southern Christian conservatives.