Turnout is extremely important for a horse's overall health! Walking around is important for their digestive system; it was designed that way! Additionally, being cooped up in a barn can cause respiratory issues and stiffness. Aside from the medical reasons, plenty of turnout is vital to their mental health as well. However, access to shelter on hot or rainy days is important as well. Talk to your barn staff, and if they won't budge (and the issue is important to you), I would recommend looking for a new barn.
I know this is not advice but that's exactly what my barn is like. Including tacking up we can only ride the school horses for 45 minutes and has 4 arenas. 3 of them are shaded but the other one is not. Also, it only has 20 stalls. It doesn't even have a pasture and nothing even close to trails. I don't hate it but I wish that it had more stalls and better facilities like a trail or pasture.