How can a private independent caregiver get paid from a long term home care insurance company?

The advice already given is excellent for a professional independent caregiver. You get paid for working. The person, their POA, or guardian pursues reimbursement from their insurance.

Note that the insurance may mandate working with an agency unless none are available, and will require that you comply with local licensing and minimum insurance requirements. Some do not cover work performed by specified relatives. There is a deductible period, often 90 days, before insurance starts to pay.

Care notes will be required for each day so they can confirm that the work complies with the care plan developed during the application process, addressing ADL deficiencies or cognitive impairment.

Yes, this is not your responsibility. You get paid whether or not the client gets paid by LTC insurance. As said, its up to the client to apply or their representative (POA) to do it. As Burnt said, you get paid, the client deals with getting reimbursed.

I get a monthly check from long term policy to pay home caregivers. It is up to me to pay them weekly and withhold SS and taxes. The amount the insurance pays is less than what I owe caregivers. I make up the difference from my own funds.

Are you a licensed business?

If not, it will be hard for them to collect on their LTC policy.

It is really cheap to get a business license and become a legal entity that meets the requirements to be paid by an insurance company.

However, it is up to your client or their reps to figure out what their insurance requires to payout.

Some LTC policies will pay for private caregivers. That would not be your responsibility to deal with the insurance company though.
The people who hired you are responsible for paying your wages. Whether or not they get reimbursed from an insurance policy has nothing to do with you.

DON'T WORK ONE SECOND FOR FREE!

Make sure you don't. I did in-home caregiving for almost 25 years. The last 15 of those years have been private-duty. If wages that you agree upon with whoever hired you are not paid on the agreed upon payday, you stop showing up. They need to know that if your money isn't there, you won't think twice about abandoning their "loved one". You can never give people an inch of room when it comes to payment for private caregiving work, because even the best families will take advantage. Believe me, I know.
No one works for free and this also applies to in-home caregivers.
Accept a personal check, a cashier's check, a money order, or even cash until the business with the insurance policy gets taken care of. Otherwise don't show up anymore.
If your money isn't there, neither are you. This has to be YOUR policy.

How have you been getting paid? Or have you?
The person that hired you is responsible for paying you. HOW they get the money to pay you is up to them.
If your services are covered under their insurance they are the ones that have to begin the process of filing the paperwork so you get paid.
Find out what the process is that you have to follow in order to get paid.
If you have been working without pay it might be very difficult to get the back pay you are owed.

You didn't discuss this before starting to work? Ask the person who hired you how you are going to be paid. It should be up to them to contact the insurance company.