@weezee, apologies for the delay in responding - we have just got back from holiday. I eventually got a response (see below) from my neighbourhood team.
"Dear Mr DavidsonsDave,
Please accept my apologies for not responding to you sooner.
At this time the City of Edinburgh Council does not have a process to promote ‘Playing Out’. As we have had several requests now to assist residents in developing ‘Playing Out’ for their street, an initial meeting has been held with my colleagues Citywide to identify the need for a formal process to be developed.
I can confirm that a wider meeting with our Area Roads Managers, Senior Play Development Officer, Legal Team and Roads Events Team has been organised to establish a formal process.
In the meantime the current process to close a street is to apply to our Road Services Events Team. I have attached an application form for your information.
Once a formal process has been developed I will ensure you are provided with details.
Kind regards,"
If anyone wants a copy of the road closure application form then please pm me. I can't find contact details of the Roads Events Team and the application form mentions guidance criteria for Road Services charging policy which I also cannot find on the council website. I can also provide these once I get these.
I still haven't actually managed to chat to Thomas from Dad's Rock about this but will keep you updated. From the above Herald article, it seems that he has also been having problems with the current system. Hopefully if more of us badger the council with emails and requests, they will be more inclined to get the formal process set up.