The existing policy does not simply continue
A homeowners policy insures a named person at a residence. When that person dies, the cover becomes limited and time-bound, and the house frequently sits unoccupied while an estate is settled. That combination, uncertain cover and an empty building, is the worst position a property can be in.
Vacancy starts counting immediately
Nobody living there means the vacancy clause begins running, and vandalism and water damage are the first exclusions to apply. Probate takes months. Those two timelines do not care about each other, and the gap is where inherited houses get badly damaged.
Insurable interest has to be established
You can only insure property you have a stake in. During probate that stake may sit with the estate rather than with you personally, which affects who the policy should name. It is solvable, and it needs doing deliberately rather than by leaving the old policy to run.