What exactly do insurance companies do?

So, my company decided to switch to an HSA plan this year, and it couldn't have come at a worse time. In case you are not familiar with them, they are a lower cost option to the rising cost of health insurance. From what I am learning, they also don't seem to save the insured any money, result in very high deductibles that must be met before receiving benefits, and require medical billing specialist skills in order to decipher who should and can be paid. Did I mention that in addition to paying the same out-of-paycheck expenses, I also now contribute nearly 100 dollars per pay period into my HSA account. A real delight. My insurance costs 3x what it did last year, and my service basically sucks.
The HSA account is where I save money, tax free, to keep paying these never-ending bills with. The reality is that the bills come in the mail much faster than the funds accumulate in the account. It is really frustrating.
After making a call to my doctor's office to inquire why, after making several payments to them, I was reflecting a 90 day past due balance, I am told that is because they applied my payment to the services from a week ago instead.
Why? I have no effing idea. All I know is that after nearly an hour on the phone with the billing person, I still didn't understand but had to hang up because I had other things that needed to be done that day. She assured me I was fine and would still get treatment regardless, and that was enough for me to finally hang up.
The people that take my blood charge the most. That's some irony for you.
I owe those people a deposit and first month's rent. It's enough to make you want to cry.
The billing person at the doctor's office said that the insurance company has denied my claims so far, but sent them the amount they were allowed to bill me. In other words, they want me to hit my deductible first. This is what I have already been told. What I then think to myself is what exactly do people at the insurance companies do, these days? I can tell you what they don't do. Provide any pleasing level of service at a reasonable cost. They also love to not handle problems with billing disputes. Hospital says you owe 10,000 dollars, insurance company says you owe 5,000, hospital says 10,000, insurance says 5,000, hospital comes after you saying they will try to collect the 5,000 that insurance doesn't pay, insurance says they only owe 5,000. Insurance sucks.
So what do they do? Other than selling insurance and collecting money, I can't think of much.
This new swing toward the HSA has me thinking, what a great idea. Bill more, do less. Charge me to save my own money to pay my own bills.
Damn, insurance companies might just be the new banks.