If the decessed had a will and it's a typical uncontested estate, there isn't a need for a lawyer, at least not here in NJ. The exectutor can go over to the county surrogate's office, file some simple paperwork that they will help you with and then the executor can pay off the bills, distribute the assets, pay any taxes, etc. It's not rocket science. I did it on two estates and it saved me many thousands on lawyer fees. The fees beyond that are just a few bucks. And the typical lawyer shyster takes a cut based on the value of the estate, not the actual time/work involved. The work is minimal. It doesn't take any longer to deal with a bank account that has $200K in it than it does for one that has $2K.
As for the original question, it seems the family has limited resources. So, while what they are doing seems unusual, I see nothing wrong with it. It seems like the cost would be a hardship on them and while they could probably bear the cost themselves, they might have to pay it off over time, etc. This way it would help them.
And I would trust the typical funeral home to do the right thing. Yeah, they could cheat you. But one thing preventing them is that most people are going to send checks. And the funeral homes always send evidence of who sent flowers, donations, etc to the family so that they can send thank you cards. Any funeral home folks know that if they tried to pocket the money, Aunt Edith doesn't get a thank you note, and decides to start talking about it to other relatives, they didn't get a note, but Uncle Eddy who sent flowers did, etc, they could be easily found out to be thieves.