A fish rots from the head, apparently. I have no idea if this is fact or a fiction that makes for a brutal metaphor, one that I heard again and again in parliament last week. The fish is parliament and the head is the Speaker, John Bercow.

Dame Laura Cox’s report into bullying and harassment had no need for putrefying metaphors. The language was clear, the message simple. In parliament, there are “systemic or institutional failings and a collective ethos in the house that have, over the years, enabled the underlying culture to develop and to persist”.

To make this report all about John Bercow is to entirely miss the point, but to ignore his role in it would also be remiss. Some MPs called for Bercow’s resignation from a position of care for those hurt and a genuine desire to repair the damage being done to the reputation of parliament. Publicly, these people were few in number.

Of course, many MPs hate John Bercow and wanted rid of him and used the report as their opportunity. They care as much about changing the culture in parliament for the staff as much as Tommy Robinson cares about sexual exploitation victims. The staff who have been bullied, harassed, silenced and ignored are just a toy for them to play with for political and tribal ends. Into the bin with them.

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The other pervading response was that we are already in a massive mess constitutionally and we need a firm hand to steer the ship. We will call this the Brexit defence. Unlike the shameless proxy war of the Bercow scalp brigade, I do think in the most part this defence of Bercow came from a place of good faith, although I don’t agree. There is a genuine concern that we need a Speaker who will not genuflect to the executive and let them get away with preventing parliament from having a proper say. Who could disagree? After all, the government is allowing a meaningful vote on the Brexit deal only because Gina Miller took it to court.

The Brexit defence runs easily with those who think that Bercow has been a great Speaker for MPs. This is an undeniable fact. He has also been a reformer and a progressive who has acted to improve diversity and with it the effectiveness of parliamentarians and also staff.

I am not in any of the Bercow defence groups. I will not defend him or defend his staying in his post. It won’t kill me, but that sentence was hard to write because I like him a lot and I like what he does. He knows how to speak truth to power, which is rare. You wouldn’t know it from modern-day online discourse but people can be good at some things and bad at others.

To me, Bercow is exactly that, but his good in no way absolves him.

The reality is that parliament’s bullying and hierarchical culture will never change until a robust, transparent and accountable process determines if someone stays or goes, not who they are mates with or which party they are from or, these days, if they are Leavers or Remainers. Power and patronage rules above all else in parliament, now more than ever. I have no hope that any substantial change will result from this report if MPs are the ones who steer what that change will be. It must be independent of our friendships, loyalties and fears.

Laura Cox cannot name names, but you don’t have to be Columbo to work out what she is saying in the report’s closing statement: “It may be that some individuals will want to think very carefully about whether they are the right people to press the reset button and to do what is required to deliver that change in the best interests of the house.”

I agree. Bercow reset and revived parliament in so many ways but I don’t think he can reset this. I just hope our fish doesn’t end up with a head more rotten than the last.

• Jess Phillips has been the Labour MP for Birmingham Yardley since 2015