Companies are often sold a rewrite because the current system looks old. Old is not the same as dangerous. A system that is ugly, documented, and understood by two people who still work there may be less risky than a greenfield project that has to rediscover every edge case the old one already survived.
Leave it alone
If the software is stable, the business is not blocked, and change would cost more than the pain, do not start a programme. Spend the attention on a different problem.
Modernize
If the system is the right shape but the wrong condition — slow, untested, hard to deploy, running on infrastructure nobody wants to touch — repair it in place. Stabilize, then improve the parts that hurt. This is usually the adult path.
Replace
If the system encodes a business that no longer exists, or every change is a negotiation with a dead technology, replacement can be honest. Do it in stages. Keep the operation running. Do not confuse a new interface with a new capability.
Simpness Solution will recommend the smallest honest option. That is not always the most exciting one. It is usually the one the business can live with.