Yeah The Don was as focused and single-minded a competitor as ever there was - as evidenced by his record. Lots of his team mates, famously Miller, found his attitude of win-at-all-costs and grind your already beaten opponents even further into the ground, fairly distasteful and unpleasant (but ironically, its an attitude that would have seemed pretty mild you'd think in comparison to how Australian teams under Border, Waugh and Ponting have played). Miller, remember, had spent many years in England, playing cricket, during the war, and was not able to consider them the "enemy" in the way Bradman and Hammond regarded each other. He also thought that cricket was in the final equation a game, and that having nearly died at least three times during the war, he, along with other ex-servicemen, found it ludicrous to be asked to treat Test cricket as warfare.
Regarding the Protestant-Catholic split and the O'Reilly-Fingleton faction versus Bradman, tbh, I've always thought it said more about Fingleton et al, than it did about Bradman.
I think lots of players who played with Don got on with him just fine, but they probably weren't as a rule, as proliferate in writing and publicizing their opinions as Fingleton and O'Reilly. I know Lindwall for one said that he found Bradman to be a very considerate leader and a really astute captain.