Oh good lord, can you imagine the shitstorm of martyrdom that would follow if we did that?
"Christians are being persecuted! Persecuted! It's a War on Christianity by <insert bogeyman here>."
For activists of a certain stripe it would be like all their Christmasses had come at once – including for many who weren't necessarily anti-abortion in the first place but would never miss a good bit of fake martyrdom.
And TBF, I suspect it might genuinely be into dodgy legal territory re interfering with freedom of religion.
I think ArabellaScott has it right, ditto Sturgeon in that BBC story linked above, that protests belong outside parliament, not outside clinics. And also not outside churches.