Welcome back!
It's interesting that you've been gone for 10 years and my bitcointalk account is just celebrating its 10 years anniversary
Should I just sweep them and start again?
I'm not sure what is the logic behind this, but imho it's wrong.
If you have those addresses' private keys you are most probably still just fine.
And if you were in bitcoin so many years ago I hope that you know enough about computers to make yourself a Tails USB, start it up without internet (and no persistency!), open Electrum from there, enter your private key and see if the address it shows is the one you expect. Of course, this is only if you want to check for your peace of mind. And this will mean that the address(es) is/are fine and you don't have to sweep, which would mean fees (and since Multibit is out of order for so many years you cannot use it for sending coins away anyway, so you have to make your wallet off private key anyways first).
The things are more complicated mostly because if nowadays you are just a tad careless about your private keys you may end up with your coins gone forever.
So... what I would like to know is, what is the easiest way (if any) to continue to top up my kids 1BTC wallets without losing half to fees?
I would first make sure the old wallets are still OK (see above) and fund the same addresses.
Just you'll have to be careful on when and where you buy those coins and what amounts you withdraw, since on-chain fees are high some of the days (it's getting somewhat better now), but exchanges' withdrawal fees are awfully high. There maybe others can help suggesting where it's better to buy from and withdraw nowadays.