How would the average family even know this was a possibility? Depending on who's making the arrangements, they may not know the person was a vet - some vets don't talk about it with their kids or grandkids. Whoever is making the decisions is usually under some stress and isn't going to want to start searching through websites to see if such a program exists and if they're eligible for it.
In order to have the costs paid directly by the Last Post Fund, you have to contact them before you make the funeral arrangements. If you don't, you have to pay the costs yourself and then apply retroactively to have the costs reimbursed - you have a year to do that. Funerals usually take place within a few days of death and most families make arrangements with the funeral home and cemetery or crematorium in the day or two after the death. None of those places will proceed without a finalized payment arrangement in hand.
Eligibility depends on the size of the estate - how quickly would you know that?