Yes, that would be very irritating for the police department and was probably unwise.
I recall there was a similar attitude of deep suspicion against police by the family of the Sherman's, murdered in Toronto. The children (who were obvious suspects because they inherited) immediately lawyered up and accused the police of incompetence, hired their own pathologist because they didn't trust the police one, tried to take over the investigation with their own set of PI's, demanded the involvement of the FBI, set up their own tip line with a massive reward.
However, police never said or leaked anything publicly to criticize them, or cast suspicion on them. Eventually their outrage apparently subsided and they seem to have given up trying to run the investigation. I don't think most people hold it against them, I don't regard it as proving their guilt.
I don't know what I would do if my child were murdered in my home, I expect I would behave extremely irrationally. The fact that people won't even consider the possibility they were acting out in grief and rage at their loss, is very interesting to me.
JMO