They take the attitude that if someone says they have been sexually assaulted, the right thing to do is believe them. Because the overwhelming odds, are that they are telling the truth.
See, you're doing it exactly there. There is a big difference between disbelieving them and actively believing and therefore assuming that the accused is guilty without trial, which is basically what you are advocating. It's incredibly damaging for the argument that anonymity to not be granted to those accused (which I really hope it isn't) when you have so many people and high profile organisations prepared to nail their colours publicly to the mast to say they believe the offender guilty before a shred of evidence is put before a court.
Believing victims doesn't automatically mean condemning people without a trial.
Most of the time it does. When a specific attacker is named or there is DNA evidence, most of the time it means that unequivocally believing the victim has to mean that you also believe in the guilt of the accused with no trial and possibly even after being found innocent. Can you name any circumstances where this would not be the case? This allows people who want anonimity for those accused to point at this and say 'If you name me, even if I am not charged or convicted, my name will always be tarnished. There are lots of people out there who are publicly prepared to state they believe in my guilt without trial or evidence, but on the strength of an accusation alone. Therefore I deserve anonimity.' It's very damaging to the argument for preserving identification.
If someone comes on MN and says they've been burgled, the general reaction is to believe them
Yes, and I agree in terms of giving support it should come from the position of belief. And bear in mind that on MN advice is given from the position of anonimity for both accuser and accused. Bear in mind, there are no campaigns which imply that all those accused of burglary should be assumed guilty without trial. If there were, I suspect burglars may well be demanding anonimity too.
Not sure how it's done now, but when I lived in Scandinavia there was a general agreement among the press to stick to anonymity for all accused persons, regardless of crime, until charges have been proved.
I believe some European countries already at least partially do this. I know Germany only names first names and initials, so Fritz S., 36 or Heidi G., 47. Which perhaps is a reasonable compromise as those who are victims have enough info to come forwards, those who are not directly concerned don't usually have enough info to identify. I don't think it's very satisfying and would prefer to keep full identification plus pictures as I think this offers the best chance of victims coming forward.
But really, I think these public 'mud sticks' campaigns really are a big reason for this being in jeopardy.