Yes - you can just seed on top. A wee layer of soil on it first will give it something to bed into. Dave's right though - a good scrape first will remove some moss and allow the seed to get in. If you walk over it gently afterwards, that'll help the seed get some contact too. The tree removal will certainly help in future. They cast shade, and also take up a lot of water. Not that we have to worry too much about that here, Duggie What is it they say, Dave - a handful for the ground and a handful for the pigeons?
Very detailed and by the far the best source of advice I've come across.
Given the extent of the moss there and from what I've read on lawnsmith, I'd do a light scarify in spring and heavy in autumn. Hit it twice in the year but not too enthusiastically in the spring when weed seeds could take up all your bare patches- and there could be a lot looking at that if you try and get all the moss.
There are various approaches and when I first scarified my lawn (in the Autumn) one that worked well for me was-
This gets the vast, vast majority of the moss, then fertilise and overseed.
- Apply iron sulphate
- Apply iron sulphate again
More detail here as to timings etc. as you need to wait between applications- https://www.lawnsmith.co.uk/topic/moss-control-removal/moss-removal
Being in a shady spot you won't want to cut the grass short, especially not for over winter.