The shutdown doesn't really save the government much money. The pay that is currently being withheld from exempted employees is guaranteed to be paid to them once the funding is there, and Congress has always voted in the past to send back pay to the non-exempt employees as well, and has indicated it will do so again now. Besides that, while the shutdown is occurring any services that the government is currently procuring from private industry have to be put on hold unless the service itself is exempt, but of course the contracts are already funded for the whole period of performance, so the government is effectively throwing money away on those contracts.
Really, there isn't very much money saved by doing the shutdown. Note that everything listed here could be exempt anyway and ignore the lapse in appropriations. New purchases are not being made, but will probably be made immediately after funds become available. Contracts that expire during the shutdown similarly won't be renewed until after the shutdown is over, but will still probably be renewed as normal. In theory, there might be less of an upkeep cost to the facilities while there are fewer workers using them, but I doubt there's any significant costs that can be reduced there.
Depending on the wording of the Congressional approval for back pay for non-exempt employees, the back pay could only be paid if they don't quit while they are furloughed. So, if the shut down lasted long enough, it would effectively force employees to find other work and forfeit their back pay. I'd imagine no politician wants to come out in favor of not paying government employees so long that they are forced to quit to start making money again. Since most government contracts have 1 year periods of performance, if the shutdown lasted a whole year it would run out most of the non-exempt contracts, which could save some money. This particular concept requires a very long shutdown and probably wouldn't end up saving that much money though, since many services are likely exempt (or whatever the term is for approving obligating funds for a contract when there is a lapse in appropriations).