Who ultimately pays for health care?
In the US, it's the payor for your plan that pays the providers for the claims they make. That's a bit of a circuitous answer, but it's essentially the truth. In a true commercial health insurance plan (i.e. a product from United Healthcare, etc) the insurer is the payor. The plan will have members from many employers, all paying premiums, and the plan collects those premiums into a fund to pay claims. On the other hand, some employers choose to be self-funded, which essentially means they provide funds into an account which is used for paying claims, and the plan is run by a third party administrator - the third party admin basically functions similarly to a commercial plan, but they're paying out of the employer's bank account instead of their own. Employers who self-fund may just pay the entire fund themselves, but more commonly, they may collect a premium equivalent from their employees as part of the funding. In either case, the payments are made from an account that is funded by or on behalf of the people in the plan. And - importantly - in either case, the employer is not (typically) involved in actual processing, payment, or decision making about claims. Instead, they select a plan and/or design the benefits options they want to provide, and then someone else runs it. Hence, employers are not typically experts on nuance around exactly how claims will pay for different types of medical needs, although they usually will have a general idea of what's covered for mainstream needs.
There are many additional layers, which may go out of the scope of this question. Most commercial and self funded insurance plans include additional funding mechanisms, such as reinsurance and stop-loss. These are essentially insurance for the insurance plan - they cover cases where single people in the plan have extremely large claims, in order to protect the plan itself from having to pay beyond their funding.
And, of course, there are many government-funded or government-subsidized plans - which essentially means it's us taxpayers that fund the plan. But since you have mentioned employer-provided health insurance, that is likely outside the scope of your question.
Ultimately though, your employer likely won't have any specific or direct financial impact as a result of your claims - even if they are self funded. Their plan will be underwritten and designed to accommodate paying for the benefits it provides.
You also asked,
How to confirm all this in a job offer?
This is an easier question. Get the plan documents from the potential employer. Just ask for the summary of benefits and any other documentation they have. You should not need to explain to anyone at the employer what your individual needs are, and giving such an explanation may just create drama. Instead, get the documents and read them yourself. Compare the claims you have typically submitted in the past (or those which providers have submitted) to the benefits in the plan. If you can't determine a clear answer yourself, contact the plan, not the employer. Ultimately, it's the payor (the plan or the third party admin running the plan) who adjudicates claims. A benefits clerk at an employer will almost certainly not be an expert on specific questions about specific claims, so don't even ask them. Go right to the source - call the people who will actually be processing your claims.
If you still don't get a straight answer, consider asking your providers. If you have a regular provider you use for specific services, have a conversation with their billing staff about how their claims are billed and how they would be handled based on the plan documentation you have. Billing staff may even be able to give you a literal answer because they've dealt with that insurance plan before. Or, they may be willing to call the plan and discuss. It's in their best interest to get a straight answer to this question, so use their help to get the answer. Billing staff want you to be enrolled in a plan that will cover their services.
Finally, it's important to be careful with requests or expectations to have an employer treat you differently than other employees (i.e. pay for health services which aren't covered by their health benefits). While this sort of request may seem innocuous on the surface (shouldn't the employer be able to spend money if they want to?), many employers will be highly reluctant to do so, since providing drastically different health benefits to a specific employee runs the risk of conflicting with regulatory requirements their existing plan is subject to. And, doing so also sets a dangerous and expensive precedent - what happens if other employees come forward and want specific coverage? Plus, this sort of arrangement would mean a direct financial impact to the employer - they wouldn't be able to pay for it out of their current insurance fund and they won't be able to take advantage of stop loss or other protective measures built in to their plan. Further, it would put them in the position of having to adjudicate the claims themselves, against whatever contract they offered you, which again exposes them to regulations and oversight which they probably don't want to get involved in. Ultimately, an employer may theoretically agree to compensate you for your needs outside of their regular health benefit, but the chances are slim enough that it's better to just focus on understanding the default benefits offered in their insurance plan, and shopping for an employer with benefits that meet your needs.
I am editing my answer after re-reading your question a few times, to address a concern you seem to have, based on these questions/statements you made:
Or is the employer actually paying the majority of this? Knowing how my family's health care costs impacts the bottom line of the company - or at least how they perceive that cost - may frame a salary and benefits negotiation later on
Even for a self funded plan, the cost is (at best) only indirectly impacting the employer. Self funded plans are designed and regulated in a way that effectively prevents a single high-cost person from financially impacting the employer. And if the plan is not self funded (i.e. it's a commercial plan), the link is even more indirect, since the risk is distributed across the plan's entire book of business.
Further, it would be illegal for your employer to make employment or compensation decisions based on your medical needs, regardless of whether or not your needs had a financial impact for them. They can't deny a raise, fire you, or change your benefits simply because you had a lot of claims.
And - to further put your mind at rest, speaking as someone who has spent a significant portion of my career doing plan configuration and data analysis work for one of the largest third party administrators in the country, I can tell you that while $80,000 of claims in a year may seem like a lot to you, it is likely far from being significant the plan. There are almost certainly other people on the plan with vastly more expensive needs than your child. Someone on expensive chemotherapy or other specialized treatments may be racking up that much in costs every week (or even one single treatment) much less a whole year. Your child's costs may put him in the top 5% or 10% of claimants, but the top 1% are likely at least two or three orders of magnitude more expensive. In fact, $80,000 in a year may not even be enough to kick in the plan's stop loss or reinsurance. It is, literally, under the radar. So don't get hung up on potentially impacting your employer's finances.
What's the most appropriate way to get confirmation on all medical services needed by my family without the employer wanting to rescind the offer because I'm going to cost too much? What's the right way to ask?
See above - and as a TL;DR summary, since this answer is a wall of text: just ask for the benefits paperwork, check to see if your needs are covered, and then direct any questions related to specific coverage for specific healthcare needs to the plan administrator.
The employer doesn't need to know your needs and likely isn't the best source of detailed information, as explained above. Effectively, if you called your employer and asked their benefits person a bunch of detailed questions, the likely outcome is that they would just call the plan themselves, anyways.