That no input thing can be a real bummer. My DH didn't listen when I "suggested" the gh needed better ventilation. I do the best I can with what I have, but when it gets above about 75* outside all the plants need to get moved outside, too.
Sounds like yours has a similar issue. Mine catches the afternoon sun.
In the winter, morning sunshine would help warm it more quickly.
In the summer, I hold a mass plant exodus, leaving nothing indoors.
It gets unbearably hot even with shade cloth.
No, what I have is actually a hoop house. If we have a "normal" spring I can begin using it in late March. I'm hoping to insulate it with bubble wrap and see if that can get me out there a bit earlier. I use electric heaters to deal with the overnight cold. We're having on oldfashioned winter with lots of snow and bitter temps. It's hard enough to keep myself warm, can't do the gh, too.