No, they don't, but we need to be very precise in what we're discussing
There are a couple of ways your main question can be read:
Will the new insurer cover a service that their insurance contract includes, even if I know I will need that service in advance of signing that contract?
Can I obtain approval for a surgical procedure under my current insurance, and then maintain that approval seamlessly while switching to a new plan or new insurer?
These have different answers, and some implications that are relevant.
1. Will the new insurer cover a service that their insurance contract includes, even if I know I will need that service in advance of signing that contract?
This is an unambiguous "yes", provided that the surgery is related to, or is itself, an essential medical benefit under the ACA, and the new insurance is an ACA-compliant plan. The essential benefits portions of the ACA mandate that ACA-compliant insurance plans must cover a large number of services treating a large number of medical conditions. Other portions of the ACA mandate that an insurer cannot deny you coverage or charge you a different premium for such a benefit because you happen to need it (whether you already know about it or not).
So the considerations are: i.) is this plan ACA-compliant? ii.) is this medical service an essential medical benefit?
If the answer to both of those is "yes", then the insurer will likely have to cover the surgery.
2. Can I obtain approval for a surgical procedure under my current insurance, and then maintain that approval seamlessly while switching to a new plan or new insurer?
This is an unambiguous "no". While the insurer must provide coverage for essential services, they still have their own procedures that must be satisfied before they can be compelled to pay. Your question indicates that this surgery has already been pre-authorized by your current insurer. That is worthless with respect to a different insurer which has not yet pre-authorized the surgery, and of unclear value with respect to a new insurance plan under the same insurer.
It is unlikely that the new insurer would outright deny to cover an essential benefit, but they may have different procedures involved in pre-authorizing a specific service. For example, the new insurer may want to see evidence that you have pursued non-surgical interventions first without relief while the old insurer might not have such a policy.
There are also provider networks to consider. If your current insurer (or plan) includes Facility A in their network, but the new insurer (or plan) does not, then the new insurer (or plan) is not going to pay for the surgery at Facility A (or may not pay the same rates with the same cost-sharing arrangements).
Finally, even if you use the same insurer in both instances you may not be on the same insurance plan, and that is an important factor to check. If you're currently on Parental Corp.'s Premium Health Freedom Plus plan through the insurer Health Co. (plans tend to have florid, non-descriptive names), and your own insurance would be the Deluxe Health Liberty Choice plan through Health Co., the specifics of your benefits may differ in ways you can't observe, let alone predict. It is also possible that your existing pre-authorization will expire before your new, preferred surgery date.
The insurer's responsibility with respect to your medical bills depends on the specific plan they offer (though high-level considerations, like "is this essential service covered?" will not change).
The bottom line: don't assume that you can maintain an existing permission, like a pre-approval, while switching among insurance plans or insurers.
Some other considerations:
I do not recommend assuming much about what your new plan will be like. It's definitely possible that you will find an ACA-compliant plan which offers similar coverage to what you have now, but with a lower out-of-pocket maximum. But it's just as possible, if not more so, that you will find yourself with an OOP max of $4,500 to $5,000.
While ACA-compliant plans are required to cover certain services for all applicants, they aren't required to offer the same insurance policies to everyone. If you're on your parents' insurance right now, there is a good chance that they have that policy through an employer's group policy. It is unusual for a small firm or individual (depending on your exact status as a self-employed person) to have access to the same insurance plans larger employers do, though it certainly can happen. But applying for something like an individual coverage plan is unlikely to be as favorable to you as a group plan would be (exceptions abound, but be mindful of this).
Your age is not as helpful to you as you might think. It is possible that you can get a better policy, but in the same way that you cannot be underwritten to your detriment by an insurer due to this pre-existing condition, you cannot be underwritten to your benefit due to your age. There are insurance plans that will give you some consideration as a young customer, but that leads to (4), below. I would not bet much on any individual or small-group health plan having an OOP max much below $5,000. It can absolutely happen, but I would not depend on it for financial planning, unless you already have a policy outline in hand that says so.
There are many insurance plans available that are not ACA-compliant. These are often aimed at people looking for a better deal on their insurance, typically evaluated by monthly premium costs. You will be able to find such a plan (depending somewhat on your state's regulations), but these do not need to cover essential medical benefits, nor are they bound by many other provisions of the ACA. Whether or not you want to pursue such a plan is up to you, but I recommend establishing in advance whether or not this procedure is covered and whether or not you bear similar risk under your new plan.
Things can change. There is currently an ACA-related lawsuit heading to the Supreme Court, and depending on how that case goes you might find 2021 to be a wildly different environment for health insurance and health care than 2020.
There aren't income limits for private insurance plans, though some programs that defray costs, available through ACA marketplaces, do. If you are eligible to participate in a plan and can afford the premium, you can't make too much money to qualify.