Yes, they "should", but not a must. The topic can be delineated into two:
1. Financing process
2. Legal (lien)
Conceptually, the quality of service is different when entities (e.g. dealers) conduct their business internally (finance via Subaru/Chase) vs externally (a lender). The outcome will be different. And you'll be treated differently if Subaru lost profit on the car sale and they are not allow to make profit on financing - worst case scenario for dealer, particularly on a bare-bone car. In this case, the OP has two choices to finance internally or externally. To implicate that the buyer doesn't have to take part in handling the financing process between the two is quite irresponsible. The advantage of financing internally is to make the deal in one sweep for easy processing. When we seek for outside lender, it's a best practice to know their process and take control. I'm not trying to preach, just a recommendation based on my experience.
The lien issue is a state-by-state cases because it's a legal matter. Conceptually, the lender (not dealer) has the choice to place a lien on the car, so it's the lender's responsibility to do that. The dealer will be paid in full and have no rights to place a lien on someone else's property. Again these legal matters are different across the US and the practice can vary.
Just as a follow-up from the perspective here. We're a medium to high volume dealership (for a Subaru store, anyway, not compared to a Toyota store). Mistakes CAN happen with our folders, sometimes they do. We take responsibility when that happens, we've had drivers take paperwork to customers to re-sign. A check here effectively can't be lost, they run it through e-Check. We have three finance producers, a finance manager, and a clerk in the store to review everything - before it ever makes it to our corporate accounting office (and we're getting ready to add a fourth producer, due to volume).
Oklahoma is a title-issuing state, so it really doesn't matter here who or how you finance your car - you're going to get the title from the state once the paperwork is complete. And when the car is paid off, you get a lien release sent to you from the bank. Other states, the title is sent to the bank and they hold onto it until the car paid off.
But as follow-up to your original topic, if a mitsake (!) has been made by the dealership, give them a chance to correct it. If they are unwilling or unable to do so, then contact SOA like Robert said. They may or may not be able to help with your particular issue - and if you're in SNE or SDC territory, SOA probably can't do anything anyway.