Earlier quoted context omitted.
Why is the conflict section in there at all, though? To the extent it's there now, it's conceding something clients hope to get through redlines; game-theoretically, it's pure negative for the contractor.
It's obviously for the client's benefit. But additional turns of the agreement aren't good for anyone. Clients typically have the leverage to get this. I've included a fairly contractor-leaning formulation, which requires the client to act.
They also don’t want the contractor to disclose anything about their business or the project to anyone else, which is why they’ll never expect you to provide a competing clients list. If they ever start to inquire, that line of questioning is trivially shut down by pointing this out. You take confidentiality seriously, and that applies to them as well.
That aside, it’s a ridiculous proposition. They are hiring you as an expert in this domain because of your experience. If you’d signed such an agreement with the first company to suggest that, you would not be able to offer your services to the client.