As for keeping their son's illness a secret, I don't think that is so out of the ordinary.
I don't either, at least as it applies to the period soon after Alexei's birth. But the situation was very different after, say, Spala. For one thing, Alexei had to be carried in public appearances. People nearer to the throne were aware that
something was seriously wrong. His obvious illness was the source of much public and diplomatic speculation.
More importantly, Alexandra's dependence on Rasputin was moving from becoming a source of comment to a source of vicious rumors about Alexandra's personal conduct and Nicholas' inability to deal with it. As it began to cut away at the reputation of the monarchy, the choice became one of trying to conceal the cause of a very visible illness or tolerating the widespread and hugely damaging misunderstanding about why Rasputin was admitted to the inner circle.
At this juncture, I suspect concealing the
cause of the illness had more to do with protecting Alexandra than with protecting Alexei. People would not have blamed Alexei for having hemophilia. Many, whether justifiably or not, would have blamed Alexandra for it.
Someone posted earlier, can't remember who, that the thought Nicholas should have listened to his parents and refrained from marrying Alex. I have thought that at times too. But all that did was probably push them together. They were, after all, young at one time too.
Alexander III was young, too, when he abandoned his romance to marry his brother's fiancee. Russian history was rife with tsars who married for state reasons instead of for love -- and with tsarinas who refrained from marrying (at least publicly) for state reasons. This particular piece of self-denial came with the job.
To get back closer to the original topic of why other Romanovs were Alexandra's enemies, I think we have to remember the various objections Nicholas and Alexandra raised to marriages elsewhere in the family for the very reason that they damaged the interests of the dynasty. Dmitri and Maria Pavlovna, for instance, were removed from their father's care and sent to Ella and Serge for raising when their father was exiled for an inappropriate marriage. Yet when it came to Nicholas and Alexandra, the rule seems to have been "follow where love leads, the consequences be damned."
I agree with all the other posters who have said that an illness such as hemophilia was viewed as something akin to a character flaw in that era and was something that a family would naturally try to keep as private as possible. That very secrecy would make it an unlikely topic to have been captured in written sources from the era. It's one of those areas that, I fear, will always be the subject of speculation and dissent.
But some things are known. We know that hemophilia, though misunderstood clinically in that era, was accurately understood to run in families from female carriers to male offspring. We know that the hemophilia in the British royal family was reported in medical journals as early as 1868. In 1873, it was confirmed that Alexandra's mother Alice was a carrier. We know that all members of the imperial family had private physicians drawn from the most qualified ranks of the profession. We know at least some of those physicians corresponded with their colleagues in Britain and Germany where the hemophilia of the British royal family was well known.
I have no means to prove it, but I cannot imagine Alexander III and Marie Feodorovna would not have been apprised of the risk of hemophilia in this marriage and would not have raised the issue privately -- and far off any sort of written record -- with Nicholas.