This is an incredibly common clause – and it simply comes from a desire for finality.
The concern is not that you’ll want to come back to work – but if you apply or are hired at a company and there is some adverse action (as simple as the application being rejected) there’s the potential that you can claim discrimination of some sort. You may also be asked to waive certain claims against the company for discrimination as part of this. These waivers don’t protect the company if there is a future adverse employment action against you even though you’re not an employee of the company – that would be a new and separate incident. The “don’t darken our doorstep” clauses (as they’re so colorfully called) are simply an assurance against future litigation.
As they are standard, the fact that they’re there shouldn’t raise any suspicion – they’re merely CYA provisions.
And any severance you get would be in exchange for this agreement. A week seems insanely short – and I don’t see why you’d be asked to sign something in exchange – it seems more like something they’d throw out as a courtesy. Generally, I’ve seen severance be based on a 1 month per year of service as a starting point.
I would also make sure that all and any communications are in writing from this point on, certified mail and email would be nice. If you speak on the phone, follow it up with the writing.
There is nothing wrong with negotiating for a larger severance. Since it all seems amicable at this point (as much as it can be), remember that the release represents a benefit to the company…you are being paid for the promise to refrain from doing things in the future that it is your legal right to do.