DEALERS PERSPECTIVE:
Never go to a bank and get a loan to buy a car!! The bank borrows their money from the Federal govt. and loans it to you at a higher rate than they borrow it for in order for them to make money. They also lend it to dealers indirectly and mark up the rates as well but not as much as they would if you walked in the door and asked for a loan. The key is that dealers can give you a better rate from the same bank because its our discretion on how much we mark up the rate to the customer. That means that if I like you, you get a better rate.
Another important aspect is that an approval is not a guarantee, many approvals have stipulations such as proof of income, proof of residence (so they can repo the car if necesary), references, caps on how high a payment they will approve etc. The biggest culprit of this is the online bank approval, the thing that looks like a check but isn’t. Capital One started this a few years ago and has done very wel because of it. WE DO NOT ACCEPT THEM AT OUR STORES. PERIOD. They are a Bank Draft, not a check and have many stipulations so do not be fooled by them because you could screw yourself
What goes into the equation:
Everything Nikkuk said in the OP is absolutely true except for the average credit score is much lower than 720, its more like 670.
Just remember that dealers can negotiate the terms of the loan with the bank on most occasions so we can be very flexible with the consumer. That does not mean that if you haven’t paid a bill on time in 3 years and owe 10,000 more on your car than what its worth your going to get approved.