For trade-in, the website makes you pay the whole amount and then (presumably) mail back your trade-in for credit.
If I were to walk into a real Apple Store, could I just hand them my phone and handle the whole trade-in (and get the trade-in credit) in person?
Yes. I’ve done trade-ins in several stores in the EU and the UK over the years (as there’s no Apple Store where I live, but even premium resellers do it this way as well). The way they technically do this is they do the trade-in first and issue a voucher (I actually got a physical Apple gift card with the value of the trade in each time), which you then use towards a new phone (or any other device) when paying. The other benefit of trading in devices at an Apple Store is that it seems they are more lenient with the condition of your device when you trade in in person vs mailing it in. They don’t scrutinise them much as they want to make a sale.
I have an iPhone 11, and all the website even asks me is if it powers on. I’m guessing they don’t care that much about the 4-year-old phones. 
Yes, I ordered an iPhone 11 and traded in my 6S using the Apple Store app on my phone. Then drove to the Apple Store and completed the trade.
A caveat about mailing in your old iPhone:
A couple of years ago when the iPhone 13 came out my wife and I purchased them through the Apple website and traded-in our old phones. When we received the shipping boxes, we packaged the old phones (separately) and shipped them via Fedex. Maybe a week later my wife gets an email stating that the condition of the phone was as stated and she maintains full credit. I received an email a few days after this stating that they never received my phone. I went on the Fedex website and see that it was delivered on the same date that they delivered my wife’s phone. I called Apple and stated that I have proof of delivery. The support person said that she would check on this. Later that afternoon I got a confirmation email stating that they received the phone and everything was fine. Moral - go to the Apple Store for trade-in’s.
Fast forward 2 years. This fall my wife and I purchased iPhone 15’s through the Apple website and traded-in our old phones (I know…but I’m thinking “What are the chances…”). When we received the shipping boxes, we packaged the old phones (separately) and shipped them via Fedex. Maybe a week later my wife gets an email stating that the condition of the phone was as stated and she maintains full credit. So I checked the FedEx website to see the status of my delivery. It showed the last location of the package to be the city where the vendor for trade-in’s is located. So, I’m thinking, surely it will be delivered by the next morning. I check the website the next day and it’s still at the same location. I call FedEx and the support person told me that usually means there were no contents in the box and suggested that I file a formal claim. Instead, I contact Apple and explain the situation. The support person said she’d check into the matter and would get back to me. Later that afternoon, I get an email from Apple that everything was fine and that I would maintain full credit. After searching on the internet, I find that this is fairly common (iPhones stolen at FedEx). I assume this is why this was resolved so quickly. Moral - go to the Apple Store for trade-in’s.
Do the store trade in if possible. It is a “one and done” transaction.
Posting back after my in-store trade-in experience. Everything went very smoothly, apart from some minor hiccups with the new iPhone needing a software restore due to a glitch of some sort.
But they basically did the whole transaction, then parked me at one of the tables in the back of the store to handle any setup that needed to be done. They let me use both the old and new iPhone at the same time, so I could double-check important apps, make sure everything worked, etc. Then when I was done they had me reset my old iPhone before they took it.
The process - from walking in the store to walking out with my new phone - probably took a bit over an hour, but that was the sales process, the device-to-device transfer, and me goofing around with various apps and making sure the important stuff was in place.
This is definitely the way to go, IMHO.
This is the exact same process I was given in my Apple Store some years back when I bought what is still my current phone, an iPhone XS! So Apple has been providing wonderful service for a while.
I’m also still using an XS (Max in my case). Mine is running iOS 17 just fine, but this is likely the last year we’ll be able to install the latest version of iOS on them.
Good to hear the iPhone XS handles iOS 17. I have not upgraded yet from iOS 16. The XS has been a great phone. Beautiful screen, battery still holding up, and no performance issues as the years have passed.
I have an iPhone 11 and can’t remember being impressed by a new version of iOS since I bought it. Most have been eye candy and “privacy upgrades”. And most recently, an address book that I absolutely hate.
Someone recently reported that the main reason users of 3+ year old iPhones gave for upgrading was “dropped my phone”. I can understand that, but I’ll probably upgrade in the next two years. Or earlier if Apple comes up with some impressive, non camera, hardware feature.
My 11 was running slow on iOS 17 for some reason. Lots of jankiness. But I figured that getting four years out of a phone was a pretty good deal. Especially since they gave me $200 for it on a trade-in, and the base model of the 15 comes with 128 GB of storage – so I didn’t have to upgrade that. 
I believe you made it just under the wire. My understanding is that my iPhone XS, released the year before your iPhone 11, no longer has trade-in value with Apple.
My carrier would probably give me some nominal amount for mine if I got a new one from them. But I’d keep it anyway. I’d hate to have my main phone break a month or two before a new model release and be forced to buy a replacement immediately because I didn’t have a backup phone.
I tend to keep things as long as they are reliable and not expensive to maintain. I’ve only had one iPhone that still had any trade in value when I finally upgraded. But my old 4S was good enough to let my brother wait for the iPhone 6 when he broke his phone.
Keeping a working phone around as a spare is a good idea.