React 15 is six years old

Just over 4 years actually, but age doesn't mean much. Plenty of orgs swear by C++98, and that's over 20 years old. I'm sure you can find many that use older versions, and have no plans on upgrading.

So long as the tool is built well, age isn't an issue. React 15 still works just fine, and, as it's just a small piece of an app, you can upgrade everything else and keep it up-to-date just fine.

and missing doesn't have hooks.

That's not really an issue. Hooks have a lot of problems, and there's many reasons to avoid them. Not having them around might not be a negative.

Given there are no significant breaking changes to go to 16 and 15

You're not factoring in the cost, because yes, there are some huge breaking changes.

a team still using 15 is career death.

Not at all. Maybe if you're a junior, not looking to stay long, using 15 could be less than attractive. But in the real world, for devs with seniority, maintaining an app well over the years is far more important. It's also just JS. You can pick up the modern stuff pretty quickly. The ecosystem doesn't move fast, contrary to what many will have you believe.

Very few actively developing teams will be using something that old.

Etsy just migrated from React v15.6.2 to Preact X earlier this year, if you want an example (they've written plenty of excellent blog posts about it too).

You sound pretty new, so some advice: age isn't a problem in the real world. What is a problem is spending your employer's time upgrading versions just to upgrade. Unless your team gets a tangible benefit from doing so, you should avoid it. You're just costing your employer money, and this isn't something that will be rewarded long-term.