"Don't pave cowpaths" is relevant here.

If the old software is convoluted, it may be because the old business processes are convoluted.

A software rewrite should be done in the context of a business process redesign. Consider how information flows through the organization. Do you really need to do all the things you're doing now? Should different people or departments get different information? Should the product you're selling or the service you're providing change? Should parts of the process be outsourced or insourced?

If an IT department is considering a big rewrite, but doesn't have the authority to look at the business as a whole, then the project is being managed at too low a level. Rewrites make the most sense in the context of a change in the underlying business.