Pershing is a respected custodian and there are no concerns about Pershing taking/misappropriating your grandfather's money.
However, the advisor and the broker could withdraw money from your grandfather's account if he signed paperwork authorizing them to do so. The only way to be certain is to review copies of the documents your GF signed when opening this account.
The most basic type of account is a non-discretionary account. This only allows the advisor/broker to place a buy or sell order for securities and only after getting your GF's permission. Only your GF can withdraw money from the account.
Limited Discretion is the next level; the main difference is that the advisor/broker can place buy/sell orders without getting your GF's permission. This is common. Only your GF can withdraw money from the account.
Full Discretion is very similar to being a Power of Attorney(POA); it's likely your GF would have signed a document providing limited POA authority to the advisor/broker. This gives them the ability to withdraw funds from the account. This would typically be done to purchase real estate or other non-tradable securities.
Pershing will require a written letter of instruction/authorization to send money to an address other than your GF's address of record, or your GF's bank. However, if your GF granted Full Discretion it is possible for an unethical person to send money to a 3rd party address or bank.
All advisors and brokers in the US are required to be registered and their disciplinary history can be found here: https://brokercheck.finra.org/
Check Uruguay's Securities Regulator for something similar - I assume they are required to be registered there as well but only know how US and EU securities markets work. I found the below site but as an english only reader I can't say with certainty how helpful this will be.