Who administers the will of someone who dies intestate, who has no
family?
There is a position known as the "public administrator" paid mostly from fees charged to probate estates in most jurisdictions. In the absence of such a post, creditors of the estate or beneficiaries of the estate can apply to do so when there is no family.
Other than the administrator, who else did he need to convince the
will was genuine?
The court with probate jurisdiction.
By what credible set of occurrences could have have led him to ‘find’
the ‘lost’ will as a non-family member, nor friend?
Is there somewhere else he could have ‘found’ the will that would have
been credible?
A companion had it. A bank had it. A former lawyer of the person had it. A military colleague had it. A landlord or storage unit company found it.
Note also that the probate court sees only the will, and doesn't hear the whole story of how it was found, in the absence of a will contest or a will not executed with the usual formalities as these wills would have appeared on their face to have.
And, if the notices to disinherited family members are not actually delivered (not mailed or mailed to bad addresses), the disinherited family doesn't show up to contest the matter until the finder's fee has been collected, the fraudster is nowhere to be found, and the charity has received the money and is being asked to refund it.
Would he have to have somehow had access to the deceased's property
after their death?
No. Often a will in not stored with someone's other property. And, in this case, he already knew that there was no actual will to find. He just needed to know that they died and weren't paupers.
How could he have gotten away with listing ‘fake’ witnesses? (doesn't
the administrator do research?)
If the witnesses signatures are notarized, the witnesses aren't interrogated unless there is a contest. Frequently, they are law office or bank employees who witness many wills and may not have a very specific recollection. The notary's authenticity wouldn't usually be checked unless there was a dispute as to the will's authenticity raised by some interested party.
Once he had convinced the right people/organisations, who would have
been responsible for selling the deceased’s assets, and paying the
money to the charities in the will?
The executor of the estate appointed by the court. Sometimes distant family, sometimes a creditor or a representative of a charitable beneficiary, sometimes the public administrator.
What benefit could his approach have had over simply acting as a
normal heir hunter, e.g. using genealogy research databases to track
down distant family members (who eventually brought his fraud to
light), and just charging them a finders fee as he did the charities?
Family, even distant family, is likely to know more about the decedent and to research that relative (previously unknown or little known to them) than a charity that never met the decedent and doesn't want to contest a bequest to itself.
Also, close disinherited family will usually find out eventually. A finder's fee recipient can be paid and gone by then. A recipient of the inheritance under a forged will can be forced to disgorge it to the rightful heirs. Money laundering of the finder's fees received would be a necessary component of the scheme if it is to be successful.
This was a pretty successful "long con". After the fraud was discovered and proved:
The relatives we had found were then able to inherit, following the
rules of intestacy laid down by law. The criminal was not prosecuted
in this case because of his advanced age.
In fraud, if they only catch you after you are dead, it is a win for the fraudster.
Footnote
In cases where the decedent had no family, this is a "victimless" crime that gives money to the finder and charity in lieu of an escheat of the decedent's entire estate to the government.
Second Footnote
This is not a common fact pattern at all. I've never seen it in twenty-five years of practice, I've never encountered someone who has seen this happen, I've never seen a news account or continuing education instructor account, or professor's account of this particular kind of scheme before today, and I've never seen a court opinion addressing this fact pattern. This may be the only time in the history of the world that this particular kind of fraud has occurred. More common instances of fraud involve:
A family member who feels great financial need duping other family members to get the entire estate for himself or herself.
A family member forging a will in an effort to win an internal family struggle where the testator isn't budging.
A family member forging the estate plan the decedent intended to make, but didn't, with little substantive change.
An outsider or family member winning the trust of a frail but wealthy elderly person and having that person execute a will leaving them everything.
A professional conservator or guardian for someone who has no close family (or unrelated power of attorney agent) charging excessive fees (perhaps for services not actually rendered) and making undisclosed self-dealing transactions (e.g. gifts to himself or herself, and unfair business deals with related parties).