I believe the problem with the Pledge was that it was a pledge to the "flag", which in the minds of the Watchtower leaders was akin to somehow pledging allegiance to an idol of sorts.
That recollection is not wrong, but it's not the whole story. For example, in schools many JW children were presented the option of not pledging to the flag and, instead, simply stating that they pledged allegiance to the government of the United States. These children refused this option for the same reason that so many adult JWs did the same thing: they thought it was wrong based on what Watchtower put out there for consumption for the general population of JWs. The linked article offers several examples of this.
So because of what Watchtower was teaching we had JWs refusing to pledge the exact same oath that Watchtower leaders were pledging, and not once did the leadership just come right out and say "Here is the national pledge of allegiance to the United States government, you should not have any mental reservations about pledging this oath of allegiance." Not one time did Watchtower leadership come right out and say this to the community of JWs. Not once! Instead Watchtower leadership put out that phoney-baloney alternative oath, which by itself suggested to JWs that there must be something wrong with pledging the standard national oath of allegiance.