Voluntary organisations can lift their sex restrictions if they want to. Men can be admitted into the WI. Men can be admitted into the women's race - it's just not legal to admit only a limited selection of self appointed men.
I think the reason organisations don't do this, is because they are aware that a significant number of men might join and behave in a way which is disrupting / harassing / intimidating or undignified for women. Which is the exact same thing that can occur when trans identified men join. Self declaration as trans identified, doesn't seem to reduce that risk. I have never seen any evidence that points to trans identified men as being less risky to women.
Men may as well be asked to self identify as "safe" or "nice" or "not a pervert" or "unusually bad at running for a man".
Womens participation is important. If an event or organisation is advertised as for women only, I think women do have a right not to be missold. If the women's institute wants to be open to men, they can be open to all men. Being open to "trans women" amounts to the same thing as any man can adopt this status at any time regardless of his appearance or behaviour.
For the record I would find it upsetting for a man to compete in any way within the women's park run category. It's cheating. It would ruin the event for me.