Probably the biggest downside is that the government would lose a major policy tool. If there are no corporate taxes then there's no way to incentivize companies via tax breaks.

It would also become more important to enforce taxation on corporate benefits. If a company provides a car, jet, mansion etc for an executive's personal use, then they need to put a monetary value on that usage and tax it as income. It's not that it's impossible to do that, but it's an additional hurdle for enforcement. Not only do you have to confirm that the company provided the employee X benefit, but you also have to confirm that they valued it appropriately.

Finally, it's just not good politics. It's really hard to sell a plan like this to voters. Even if it doesn't raise the taxes of the average voter, it sounds like you're moving the burden of taxation from rich companies onto the average citizen.

I think there's a lot of economists who love the idea because it really does simplify things. Besides the framing, there's not really a big difference between subsidies or tax breaks. But that third point is probably what kills it.