If an American plans to stay in London for up to 6 months, NO LONGER, do they need a visa or any expectations when showing up at UK border?

You are asking this precise question and many derivatives on multiple occasions.
The real answer to all of your important questions like this is to do the research yourself as anyone on a site like this can tell you any old bulldust just to play with you or because the answerer doesn't really know what they are on about and are just going one hearsay.
There are plenty of accurate and very informative sites that will give you all of the correct information that you might need.

You will have a Tourist Visa. Immigration will ask you how long you plan to stay, where you are staying and how much money do you have to support yourself.
UK

YOU DO NOT decide how long you can stay.

UK Immigration will give you an exit time.
The MAXIMUM allowed is six months. They can say NO and send you back where you came from.

The basic rules have not changed. You are expected to have a
REASONABLE
plan for your time in UK.
A method to fund your time in UK
Be in good health
not be a criminal
and a plan to EXIT the UK when visit is over.

Covid 19 has added a few more health restrictions.

WAVING your passport is not how it is done. You will be asked some questions and based on your answers the Immigration Official you deal with will decide NOT YOU.

Immigration Officers actually live in the UK and are familiar with the normal costs of things. MOST visitors cannot afford a six month vacation.
Up to you to explain to their satisfaction.

Technically an American can stay for up to 6 months in the UK as a tourist. 
Realistically, 6 months will be seen as a red flag. 

For TOURISTS ONLY, US citizens may be allowed to enter UK for up to 180 days. You need enough money to cover all typical tourist expenses: round-trip tickets, hotels, restaurant meals, sightseeing, local/ground transportation, etc. And you cannot work, look for work, etc. If entering UK for any other purpose than tourism, you need a visa.

I say just show up and apply for a visa when you get there, like in the airport. That way you'll find out through experience. I did this upon going to an African country and was fine to enter and get a tourist visa, although I might've had some trouble initially.

When you enter the UK you will be asked how long you plan to stay. When you say you plan to stay for a time longer than for a regular vacation (a month or so) you will be closely questioned regarding how you can afford to not work for such a long time.  A "wrong" answer will get your entry into the UK denied. 

if you want to stay for 6 months then you will need to apply for a tourist visa before going to the U.K