Yes, if a 3rd party insurer is paying for the repairs then you are more at the mercy of them. That's why it's nice to have full coverage and you can use it to fall back on to perhaps get something more to your liking.
Lienholder/lease vehicles do require a different standard of handling. Some leases will specify OEM parts only. The insurers generally have to abide by that if it's in writing. Not many do, however. If you have a lease on your car and turn it in with damage, you get charged. If you turn it in with substandard repairs, you get charged. If you have the insurer's direct repair shop fix it and it's substandard...hooray, they have to deal with it. I get those every now and again. (An aside...never, ever lease from Hahn Financial. They nitpick every car and charge you for every door ding and curb scape. Nobody leases from them twice).
The reason cars get salvage titles/totaled when we see something that's an easy fix is pure economics. We total cars at 75-80% of their normal value. On a '86 LeBaron convertible, needing a headlight and fender will do that. Right now we're junking Isuzu's pretty frequently because getting parts for them is a problem. We junked a lot of Daewoos before GM bought them. If we're footing the bill for a rental and we don't know when/if we can get a wiring harness for you Saturn Vue, we can decide it's not economically sensible to bother fixing it if we can document why.
Not a week goes by that I don't hear someone say "I can't afford to replace it for that". I'm immune to it now. If you don't carry collision on a car worth $6000 and you only have $500 in the bank, you're making bad decisions. I can't help that. If you're driving a 200,000 mile turd but aren't willing to replace it with another 200,000 mile turd because you're only getting $1200, I can't help that. If you owe $7000 at a buy here-pay here ripoff joint for a '96 Cherokee worth $3000 (yes, this has happened), sorry, I can't help you. The number of colossally bad financial decisions I see would blow your mind. Yet, I never penalize the folks who bought a super clean Crown Vic from the grandpa next door who quit driving for $500 and it's worth $5000. It's worth what it's worth.