Greenhouse under oak tree

Looking for expert opinions on my dream greenhouse placement. (photo taken at 2pm in august)

I want to put a brick bottom greenhouse at the back of the garden under a large oak tree. it’s quite shady in the peak summer probably only gets 2-3 hours of sun in the afternoon. But in the early spring and autumn when the sun is lower we get lots of sun in the late morning until late afternoon.

Its primary use will be to start veg off in the spring and have somewhere to sit and spend time when it’s chilly but sunny. The ultimate plan is to create an outdoor living space across the back.

The obvious worry is acorns and branches falling but I thought I could put a canopy over it in the autumn/winter or cover in corrugated plastic to reinforce it.

Tree roots should be too much of an issue there’s a patio that’s been there for 10 years odd and it hasn’t risen or warped.

Thoughts?