But seriously, what Larabee says - you can't just come as a visitor, run a business remotely, and decide to stay.*
[*Unless you find something in the regulations that allows it.]
You can’t under any circumstances - because it would be illegal (‘deciding to stay’ would mean becoming an illegal overstayer, and you cannot switch from a visitor visa to any other kind of visa from inside the UK).
Also, a visitor visa extension would not be granted either.
I’ve been on the forum for over 12 years and I’ve never heard of anyone getting a visitor visa extension... although the UKVI website says it’s possible, in reality, there’s pretty much no way one would actually be granted.
So, you could come to the UK for a visit, but you would have to return to the US within the allowed 6 months, you would not be allowed to carry out any work at all while in the UK, and if you were eligible for a visa to live in the UK, you would have to apply for it and receive it in the US before returning to the UK.
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