Is it ever worth paying money to acquire a domain name for your startup?

The name my cofounder and I would like to use is taken and the current owner quotes us $7000 for the domain name.

Is it ever worth for a startup to invest a tidy amount of money like that in a domain name?

The real questions are how much is a brand worth, and what priority is it. I'd suggest that a brand isn't effective until a later stage of the startup, when product-market fit has been achieved.

If you don't have the budget to know your startup's going to last through building, pivots, and starting to scale, it's probably not worth spending thousands on something you might not even get any real benefit from.

Startups can also go through a lot of changes (target market, pricing, product focus) before reaching this point, and a lot of those changes would impact what kind of brand you want.

Lots of startups change their name after they've gotten funding or reached scale with no problem. Invest in a domain when you have the money and the certainty to do so.