It might work, but the odds are against it.
The research on it shows if any alfalfa plants are present, they need to be dead, and then wait 3 months of good growing weather before replanting alfalfa seed. A growing alfalfa plant over a year old is putting out toxins up to 2 feet around it that suppress new alfalfa seedlings.
Most would plant a grass crop in between to use up the N you get for free in the ground, like corn, but technically you could kill the alfalfa after first cutting, then come back in early fall and reseed to alfalfa.
A person can get lucky and have it work out inter seeding or waiting only 2 weeks, but that will be the exception, not the rule. It works just often enough that a friend has had a friend hear of it working so it must be ok right?
If it works, likely your old alfalfa was dormant from a long drought and allowed the toxins to break down, and the old alfalfa was not actively growing to put new toxins in the ground for several months. Then you need to get lucky and have a good rain after you seed to grow the new seeds. All in all a very risky thing to try, with how expensive alfalfa seed is.
Most times he will get the same patchy poor stand he has right now.