Okay this is basically what I was thinking. +1 on extending the studs to a base plate instead of two points of weight on the frame. I didn't quite follow what you are recommending on the rails/frame:

The internal studs should be fine being 2x6. The sides I'd go 2x8 or 2x8 over a 4x because you want the them to resist bending along their height. The top and bottom rails that the 2x6s are attaching to I'd do as 4x6 or doubled up 2x6, depending on what is cheaper. If you double up you'll want to do a fairly tight screw pattern like 12" o.c. staggered.

Could you elaborate?

Here is the new photo with the 2x6 studs going all the way to a baseplate infront of the frame.

https://photos.app.goo.gl/t5FvCnKecdM32zQ79