Unfortunately, drag force is proportional to the square of velocity... So the slingshot (or railgun or whatever) maybe gives you a nice acceleration at the start, but a LOT of that initial force will bleed off as drag early in the flight in the thick thick low atmosphere.

(Consider the difference in difficulty of riding a bike at 15mph vs 16mph and 25mph vs 26mph... 25->26 requires a hell of a lot more effort, all due to drag.)