Is there any upside to contributing to the after-tax 401(k) instead of the Roth 401(k) via paycheck deduction?

The notice above the choices in your image actually says it: the after tax option is good for when you reach your annual limits with your 401(k) account. 401(k) contributions are limited to $19,500 per year. This limit is for combined contributions to both Roth and traditional 401(k).

If you are contributing the full $19,500 to your 401(k) account, and want to save even more for retirement, this is giving you the option to automatically put more of your paycheck into your company's retirement plan. You will get no tax benefit for doing so; you will pay taxes on the money you put in up-front (the same as if you had put it in your bank instead), and also pay the capital gains tax on any growth when you withdrawal it.

You could take that same money and just use it to purchase mutual funds on your own for the same effect. Doing it this way is a trade off between convenience of having it done automatically with each paycheck, but having fewer/limited options of what mutual funds can be purchased and which broker can be used.

To understand the after-tax option you have to go back to at least the 1980s. At that time there were no Roth IRA's let alone Roth 401(k)s.

When a new college graduate opened their first retirement account with their company, you had two basic choices: put money away pre-tax, but not be able to touch it without severe penalties until you are 59.5; or put money away post-tax but know that you can get your money out without the severe penalties. This decision had to be made soon after the 1987 stock market crash. The post-tax account was seen as a way to ease into a 401(k) program. Eventually I stopped contributing to the post-tax account and made all my contributions pre-tax.

My second encounter with post-tax contributions was when was putting enough money away to hit the annual maximum in the last 1990's. I discovered that you could exceed the maximum amount if you also were contributing some post-tax money. I did this because the only way to specify your contribution amount was integer percent. It was impossible to exactly hit the number on the last paycheck of the year. But if you allowed the company to put the excess the money into a post-tax sub-account you could pick a percent that you wanted, without having to worry about what happened if you received a bonus or overtime during the year.

Now these accounts aren't as good as a Roth account, because the growth will eventually be taxed as ordinary income. Most people don't even know they exist. It is even possible that some 401(k) programs don't allow them. If I had a source of good financial advice back in the 1980's I probably would have avoided the post-tax contributions. If the situation was the same in the late 1990's I probably would have still put the excess into the post-tax account.

In your question about the Vanguard interface, I see no issue. I knew exactly what the situation was. You are in the retirement area and all the money is retirement money. But if somebody doesn't realize there are three flavors of money they could be confused.

tl;dr: The only possible reason to prefer after-tax is answered by OP in another question from years ago.

To answer your title question:

Is there any upside to contributing to the after-tax 401(k) instead of the Roth 401(k) via paycheck deduction?

No*. In fact it's the opposite. It's the same concept as after tax contributions to a Traditional IRA. There is absolutely no good reason* to make after-tax contributions to a Traditional IRA or 401(k) instead of their Roth counterparts, if you are still able to contribute to a Roth. In the case of an IRA, income limits prevent some people from being able to directly contribute to a Roth IRA, so they may elect to do after-tax contributions to a Traditional IRA. In the case of a Roth 401(k) there are no income limits, so you would always* want to max out the Roth 401(k) contribution first in lieu of after-tax contributions, and then make additional after tax contributions if you wish to.

The reason having money in a Roth 401(k) is probably always* better than having it in a Traditional 401(k) as after-tax contributions, is because within the Traditional all growth is taxable, versus not taxable in a Roth. This is true for both an IRA and 401(k).

Of note is that whenever you have after-tax contributions sitting in either a Traditional IRA or 401(k), the goal should be to move that money into its Roth counterpart ASAP. As soon as you do so the growth is no longer taxable. With a Traditional IRA you always have full control and can move it anytime (perhaps after 1 day for clearing). With a 401(k) typically you can't move it until after you leave the employer, however, some employers do allow in-plan conversions from after-tax Traditional 401(k) to Roth, and if you have that option I would consider taking advantage of it as soon after the contributions as possible. Though first I'd confirm how often you are allowed to convert; if it's limited you may want to wait until you have more to convert before pulling the trigger.

The fact that you can convert after tax contributions (and its growth) to a Roth at some point in the future is one of the big advantages over just investing on your own outside of the 401(k). Even if you never converted it over to a Roth, a second, lesser advantage, is that you won't have to deal with taxable events on those funds until you retire.

*The only possible reason I know of when after-tax might be preferred, is a wrinkle found (by OP) in another question. If you were over-contributing beyond your means and you thought there might be a chance in the near future that you'd need to take that money back out, it appears that withdrawing after-tax contributions can be done without the 10% penalty, even if you are under 59.5 years of age. In a sense, you could use the after-tax contributions to a Traditional 401(k) as an emergency fund.