It sounds like you need to think a little bit more about what you want to do. Do you want to make an investment, or do you want to buy a vacation home for your own enjoyment and use rental income to offset the cost? Both are fine, however, you would use a different framework to think about them.
If you want to buy an investment property, you would more or less be property agnostic. Imagine you have two properties that perform exactly identically. They cost the same, they need the exact same amount of work, they were build in the same year, they would rent for the exact same amount, they are in school district of comparable quality, same cap rate, same cash on cash return etc. You probably wouldn’t care very much which one you buy as an investment. If you want to do turn key, you don’t care if the house is in Memphis, Little Rock, or Cincinnati. If you want to do everything yourself, long distance is obviously harder.
This is a very different case from buying a personal property that you use yourself. If you want to buy a vacation home by the beach in Florida, you want to buy a vacation home by the beach in Florida, and you care very much about what the house looks like and where it is located. Now, if you want a vacation home to use in the winter and then rent it out for the rest of the year to offset the cost, that is a reasonable idea, however, that is very different from buying a property purely as an investment.
From your question, it sounds more like you might want to do the latter. If you decide to do this, keep in mind that you probably could not use the property as a traditional rental since most leases are 12 month long. You could of course AirBnB the property, but be aware that the vacation rental business is much more akin to running a hotel than to a traditional 12 month rental. A long term tenant would never expect their landlord to provide furniture, dish soap, or toilet paper, however, you can’t really do a vacation rental and not have a bed in the unit. You’ll also need to have the unit cleaned after each guest. All of this will require much more hand holding – you need a local cleaner, you need to figure out how to give guests the keys, you need to manage the bookings etc.
Alternatively you could just take your $200k and buy a property solely as an investment and then use the investment returns to pay for your vacation in Florida. Like I said, either option is fine, you just need to decide what you want to do and only you can answer which one works best for your family.