The long and short here is the usual story. You, the taxpayer, are going to ultimately foot this bill either way. Either through taxes or tolls (though likely both).
The privatization angle certainly does a much better job of lining pockets of the upper echelons while providing little in return, and tends to appeal to silly libertarians who tend to relish the concept of paying 8+ bucks to cross a bridge, as long as they believe they're not forced to contribute to some other road they aren't currently using.
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Elon Musk, or anyone else wanting to go to mars is completely un-associated with the running of a nation.