Unlike employees, clients generally want their contractors to have experience working with competitors.
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.
You're absolutely right to connect terms about work for competing clients with terms about confidentiality. You're also right that the value of many independent consultants---including attorneys---derives directly from diverse experience within a competitive field.
However, the desire for experience with competitors doesn't exclude valid concerns about ongoing, concurrent relationships serving competitor-clients. To borrow from legal ethics, it's one problem to have information from one client that could be useful to another, and a distinguishable but related problem to face a conflict of interest, with both sides expecting undivided loyalty. Terms about work for competitors assume, practically, that confidentiality obligations will break down when the contractor works for both sides concurrently. We can't build Chinese Walls in our minds, or in small firms of a handful of people.
Back to the terms, note that "Early Termination" gives each side the power to terminate early, with a given number of day's notice, for no reason at all. The main effect of "Conflicts" is to require the contractor to notify the client of potentially concurrent work for a competitor. If the client has concerns, they can terminate immediately, instead of with n days' notice. It's fundamentally about ensuring the client the information they need to make use of their termination right.
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.