I have been on both sides of layoffs at tech companies in America and have both received severance and given severance.

I have successfully negotiated a severance package, but I had leverage and was very senior and took on the risk of hiring and paying for a lawyer. Assuming you aren't a senior employee, this probably doesn't make sense.

In your case, they generally aren't giving severance as a courtesy or a thank you. Their goal is to limit their liability and they do so by having you sign exit paperwork. That paperwork usually includes non-disparagement clauses, reiteration of non-compete clauses, etc. They need you to sign it so they can close the books on the RIF, so to speak.

The only negotiation point you generally have is your willingness to not sign the document at all. You can tell them it's not worth it for you to restrict your job search, and hope they call you and up their offer. Are you willing to have that fight over a gym membership? Seems short sighted.

All that said, if you aren't sure what your docs say or aren't happy with the offer, you should call a lawyer. Lots of apps online that can connect with a lawyer for a 30 minute chat for a reasonable price.

And for folks in America: you can negotiate severance when you get a job. Easier to say "hey, I'm taking a risk by taking this job. If I'm laid off, I want a month salary for each 3 months I'm here" or something to that effect. Especially if you are a little more senior. I watched a private equity firm try to fire a c-level executive after a firesale, and he just sat with his feet literally on the desk saying "i hope they do. they gotta pay me 2 years salary for firing me for any reason besides stealing from the company".