It isn't the job of primary schools to be for or against the 11+ and grammar schools, it is their job to teach the national curriculum. I think people mistake an absence from the process in preparing for the 11+ as an ideological objection. They may or may not have, but either way, their opinion is irrelevant.
And from a school /LA / government/ schools admission protocol, the premise of the 11+ is that it is a test that can identify intelligence and aptitude irrespective of prior education. So how could a state school be given the job of 'tutoring' . (Space for much hollow laughter)
Many teachers are not supporters of Grammars, per se. For all the reasons people generally are not, and presumably they see first hand how many children develop different areas of education at different times, have a loyalty and commitment to all their pupils and don't see the benefit of a sheep v goat divide on one test on one day.
They are teaching kids who, for example, will go into comprehensive schools with very high SATS in Maths and average in English, and those skills will be built in in appropriate sets in Secondary.
Other teachers will be pro-grammar, of course. And they will see Grammar-material kids who are not being tutored in NVR and VR from yr 3 or 4, or at all, and they may see the pushy tutoring parents as the problem as much as the others!
OP: my kids went to a primary where they were happy, enjoyed learning and were well taught. They then went to Comps as high achievers and got / are getting top academic results.
It isn't all about the Grammars.