Lender said they don't accept first time home buyer program?

We live in New Mexico and we are trying to get preauthorized for a loan, but they told us upfront that we would need the entire down payment in our bank account and they don't accept the state-funded first time home buyer loan. I will admit I don't completely understand the whole process but this seems... odd to me? Why do the first time home buyer programs exist if they can be ignored or circumvented? Are there any other options we would have to get help funding this or do we just have to wait until we have the downpayment on hand?

Edit: Thank you everyone who commented, we learned a lot from this thread. This is an awesome community! As a small update, the other lender we were in talks got back to us today and they have a much more feasible setup for us, so things are looking up.