here is some advise from a licensed damage appraiser.
to check nose "sway" w/o putting it on a frame machine close the hood just to the point of latching. check the body line on the undamaged side to see if the gap opens up. if it is tight in the rear and gets wider in the front you have some sort of nose sway(it could either be frame sway or the rad support is swayed) the frame assy is stronger than the support. also you do have rt suspension damage. the top of the wheel is in and not angled as far as the lt wheel(turning wise). this is definitely not a total(at least i would not total it. some ins companies can be funny though.
say for example your damage is 10,000 total. the vehicle is worth 30.000, the salvage value is 15.000. the ins co could do whats called a constructive total due to possible "hidden" damage and a high salvage value.
frames an be straightened as long as they are not "kinked". bent frames are repairable. i do not think 4runner frames are sectionable. trailblazers for example are hydroformed frames and GM actually makes a kit to do such a repair.
the shop is giving you a run around on not writing an appraisal. a "good" shop would write an appraisal and when the appraiser gets there the shop would give the ins co appraiser the estimate to review.
i would go with my own ins co. they would be more sensitive to your needs, in most cases a 3rd party carrier, you would be athorn in their side. your ins co would pay you and leinholder(if you have one) or you could sign a DTP(direction to pay) so the payments go directly to the shop and it bypasses the leinholder. then your ins co would subrogate against the other ins co.
good luck.
This is very helpful - thank you. I will look at it again tomorrow and see what I can determine by looking at the hood. The police report is almost ready, so it shouldn't be too much longer, but I do want my insurance company to handle this and subrogate it with State Farm. I've learned a lot here and I'm most appreciative