Thrifty is ok - not great, but ok. If you've reserved something particular, you want to call the actual location a few days in advance and confirm everything. If you're worried, do the same.
However, this is standard operating procedure for car rentals. They do not worry about deposits until you are actually at the facility. Rental companies also rarely let you prepay, if at all.
Also, you should be aware that they may not take a deposit on debit card.
Personally, I like Hertz, and I rent almost exclusively from them - about 40 times a year.
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