My guess, weatherwise, Mid june will be similar to beginning/mid of september. It can be warm to VERY hot, but also a chance it wont always be blistering hot. It still isn't july or august.
I use a very light down quilt and when its really too hot, just a liner. Won't touch the blankets in most albergues, but thats personal, i think.
Mid June will be busy, but september can also be very busy. You would know this. Seeing as you are starting at the end, it will be ok. I cant imagine you need to book ahead for any albergues.
Lots of people you'll meet probably started around mid may and may is just a bit less busy then June.
I don't think the walk to Finisterre is that busy at all, but i can only speak for periods may and sept. I dont know what happens in July and Aug
That said, i find that a lot of people, very enthusiastically
, start out saying they will walk all the way to the ocean, but in the end just a small portion of people do. And that is understandable. They arrive in Santiago and mentally this feels like "the end". Or they are just plain tired and don't feel like walking anymore. Can you blame 'm?
When i walked to Finisterre the first time, it felt very weird to me. I arrived at SdC, got my Compostela, went to mass, said my goodbyes to new friends and then.....i started walking again. This felt strange....but only for a few hours. After my first cafe con leche it felt like being back on the road again