I think we‘re conflating several things here.
AIUI, the OP wasn’t planning on complaining. It was more making the premises owner aware something had happened which could be avoided with child gates.
The type of booking is also not clear, but it sounds like the playgroup is a church initiative rather than someone paying £20 for an hour’s use of the premises. I‘d be interested in any advertising they do (St Mungo‘s Stay and Play), as I think if there had been a bad outcome, they would be seen to have some ownership of the event and some liability. And, their insurance policy will involve risk assessments too (the insurance policy will cover accidents on the premises of members of the public attending church events).
Churches are surprisingly well organised these days, with safeguarding policies, health and safety policies, etc. Exactly how this is done depends on the denomination. CofE is probably diocese-based. Baptists are church-based, following advice from Baptist House for those in the Baptist Union. Etc. Without knowing the denomination, it’s difficult to say who would be the right person to talk to, but the church will have a website which should contain this info.
Churches also want members of the public to come in. They want to share their faith, so many will offer things like a toddler group as a form of outreach- people come in and lose their fear of church. A church will therefore want the groups they facilitate to be safe spaces. Often, they will provide volunteers - who see the work as their service to God.
When DS was small I went to a British Baptist playgroup with him. A relatively large congregation, they had a paid youth worker and a number of retired ladies running the free group. The volunteers were all DBS checked.
I also went to a German village church playgroup with no church representatives. But the group had excellent knowledge management- each new member knew to secure the door and sweep up afterwards.
If the church is in any way associated with the playgroup, they need to know what happened.