Let's break it down:
many of whom identify as lesbian
Straight away there's a conflict. The protest argues you can't identify as a lesbian. A lesbian is a female homosexual. Clearly those attached to the statement don't believe so. Ergo they are immediately in conflict.
All of us are active in the arts, community organising, the media and education.
How is this relevant?
We have all benefited from the deep analysis, radical lifestyle and astonishing bravery of the lesbian feminists who came before us
"...but we're about to ignore all that in the rest of this statement."
However, these views are expressed by many different people, cis, non-binary and trans. There was almost no mention of this. So we can only read this protest as a deliberate attack on trans women.
So the authors acknowledge the abuse but it is more important to them to put the blame on the lesbians for blaming the wrong people in their eyes? Are they saying they are in a better position than the lesbians themselves to tell us who is dishing out the abuse? And again, it wasn't an attack on trans people. It was a protest against the trans activists who are abusive.
It isn’t trans women who...[followed by various examples of bad behaviour]
No but it is trans activists who are responsible for "the cotton ceiling" and all it's associated public musings on the internet:
www.mumsnet.com/Talk/womens_rights/3294339-cotton-ceiling-evidence-thread
Lesbian feminists of all genders
small group of cis lesbians
Again, statements in conflict with the definition of lesbian, at the heart of the protestors' concerns.
in order to spread the same kind of dangerous lies about Trans women.
Not dangerous lies about trans women. Truth about what certain trans activists say to lesbians. Gaslighting. It's as if these people deny that people like Riley Dennis exist.
The irony is that the protestors failed to point out these obvious and active factors in the continued, structural oppression of women, trans and queer people here and abroad.
This is after the ramblings about G4 etc. So it would have been worthwhile if they'd have protested about all of the problems with Pride?! Organise your own protest about this sort of stuff if it's such a big deal!
Instead they hijacked a moment in which a vital yet threatened public service, the NHS, on its 70th anniversary, was leading from the front.
People have marched specifically for the NHS recently. I highly doubt this sets back the NHS when it was just one component of a non-NHS specific march. Besides, what is a protest supposed to do? They have to get themselves noticed. Would the authors had been happy for them to be in front of the NHS marchers if they had been protesting "obvious and active factors in the continued, structural oppression of women, trans and queer people here and abroad"? Make your bloody minds up.
we are here to remind the protesters that you can’t pick and choose your history
Interesting turn of phrase, given the historical definition of lesbian is being picked at and history is being revised with people being retroactively trans-identified.
There are many campaigns and acts of solidarity we can be engaged in, including
Oh yes, look after everyone else's rights but don't you dare stand up for your own rights. It's just not inclusive enough. 
I would genuinely want to know if every single signature on that letter fully agrees with and would happily release the statement "a lesbian can have a penis. Any lesbian who refuses to have sex with someone with a penis is transphobic and a vagina fetishist". It would make their position a lot clearer and get rid of all the bull in between. Because a disagreement on this is what it fundamentally comes down to if you actually pay attention to the concerns of those women protesting.