It absolutely DOES matter. A gradual rewrite is much more likely to work than a stop-the-press rewrite.
The best way to do a full rewrite is incrementally, with test support and consideration for natural separation of internal subsystems.
Consider that if this were not true, then there would be no way to describe an incremental full rewrite, nor any way to describe a from-scratch replacement of a subsystem.
I've written on this topic before, for example https://increment.com/software-architecture/exit-the-haunted...
At some point you end up trying to change a pumpkin boat into an aircraft carrier, and there's no obvious way you can do that one piece at a time.
Which is why you do it in stages: add scaffolding until local rewrites are possible, then rewrite the business logic, then tear the scaffolding down.
A lot of low hanging fruit to be addressed that will likely lead to meaningful improvements. Once the code is in better shape and some unfortunate legacy pattern is identified, than it can be considered time to re-tool the architecture.
Because the migration doesn’t block new features, that means the org gets tired and reallocates the effort elsewhere before it’s ever done, with no immediate consequences. Rinse and repeat.
I recently witnessed this mess and it is an enormous mess. Don't build Ship2 in the first place. Instead, replace Ship1's mast and sails, and rudder etc until you've replaced all the parts in Ship1. That's the SoT approach.
What you're describing is a "stop-the-world" rewrite.
Edit: A full rewrite always meant replacing every part of a system. Whether you do it gradually doesn't really matter.