Even if that were the case both the lawyer (who executed the will) representing the deceased and the executor (who has already documented legal agreement to be executor) (both of whom know in advance of the client being deceased that the deceased wants a Catholic funeral) at the very least could prevent the family going ahead with the funeral until probate was granted. I don’t know where you come from but in my home town in Scotland probate happens very quickly.
The family cannot rush a funeral through contrary to the deceased wishes.
All I can say is probate may happen quickly in many jurisdictions, but if there are family issues I cannot believe it will happen quickly. To happen quickly, there must be consent from all interested parties: those named in the will and those who have a claim in intestacy. If unanimous consent cannot be had, the court will schedule a hearing with sufficient time and notice to all interested parties.
The point of notice is to give interested parties an opportunity to object to the will. This process can take weeks, months or more. The exact time frame for the initial hearing will vary from jurisdiction to jurisdiction, but I can’t imagine it being less than a month.
Perhaps Scotland has some specific statute concerning burial, but in most, if not all, states in the USA it is the next of kin who makes funeral arrangements because of this very issue and the desire to get bodies buried. As a practical matter, very few, if any, wills are probated by consent within days of death. At death there is too much angst to deal with getting consents signed, let alone finding everybody who needs to give consent. Even with probate by consent it is usually weeks before the will is probated especially if some family member has an ax to grind.
Admittedly, few people want the expense of a will contest. But few also want to take the time for an injunction action to stop the next of kin from burying the deceased before the will is probated.
A will is simply not a terribly useful instrument to address this issue.