If the private company is granted a defacto monopoly, it doesn’t matter that they’re a “private” company, they will have the same incentive and accountability problem.
What we know for certain though: Government taking over something is definitionally a monopoly and 99.99% of government employees are not subject to the accountability mechanism of elections.
Historically, the largest boondoggles of waste have always come from government, given they can legally hold a gun to your head and take 50% of everyones money to fund their “projects.” Private companies can’t take your money by force, unless being given those contracts by government. So again, the the incentive issue fundamentally arises from an entity being entitled to gather assets using violence rather than voluntary exchange.
IT is hardly something we need to do occasionally, so build up a department that can do it (not just write up huge reports about what it should do and outsource, like Logius) and invest in the people that will work there (retaining them as much as possible). Give a big middle finger to consultants, and listen to the tech experts. Build boring stuff that works instead of a new app every month.
It's not impossible in theory, and cheaper in the long run. It's impossible because asshats who would actually benefit from left and centre politics keep voting right-wing parties in to power.