Does the (non-resident alien) owner of an LLC (electing as a partnership) pay income taxes on (non-US) profits?

Some preliminary thoughts:

A single member LLC is considered a disregarded entity for US federal income tax purposes. So for such purposes, you are treated as the owner of the underlying business.

The territorial source of business income is relevant, of course, but the first question you need to ask is whether the income is effectively connected (what is normally known as "ECI") with the conduct of a US trade or business (meaning, more or less, is the business being run from the US)? In making that determination, the location of the e-commerce site servers, the number of days you spend in the US working on this, the location of the inventory (if any), etc., etc. are all relevant factors.

If the income is indeed ECI - than it is subject to US federal income tax. If it's not ECI - it's probably not subject to tax.

Which brings up the question - if the business is conducted from outside the US and its income is generated outside the US: why is it owned by a US LLC in the first place?

Trying to answer one part only of your tricky question:

First, consider a "totally normal LLC, owned by some American citizen living in Denver". So Frank Mountaintop, of Denver, owns the LLC.

We will refer to that as a POLLC. (Plain Old LLC.)

Let's also assume the profits OF Frank's POLLC are not income generated from trading in the U.S. For example an e-commerce site that has only sold to residents of Australia, and has not engaged in business with any U.S. entity.

It makes utterly no difference where or what the profits come from. It makes no difference at all if it is retail (the guy sells ski hats, on the internet or elsewhere), services (the guy makes software for ski resorts), day trading, prostitution, plumbing, crime, or anything whatsoever. It makes absolutely no difference, at all, "where or how" the income comes from.

Frank just (pretty obviously really) pays tax on it. And hence the POLLC "pays tax," since LLCs don't really exist, they are just pass-through entities.

(Aside: you may be thinking of issues relating to "sales tax", tariffs, or other similar fees, which may or may not have to be paid to Australia in the example, the US, Colorado, or who knows what. But that has no connection at all to the plain fact that POLLC "pays tax on all income" and there is no factor whatsoever regarding where or how the money comes from.)

So that's the answer to the second part of your question for a POLLC owned by a yank.

Regarding the first part of the question:

(A) I have no personal experience of this angle.

(B) The whole idea of a foreign person owning a US LLC ... I know it is posisble but I've never really understood why one would do this or what the advantage/whatever would be. Note too that if you try to "work for" (whatever that means) the US LLC, and you don't have a work visa, you'll be crucified (and you'll never get a visa in a million years using the "ha ha, I'm the owner" trick these days).

(C) It does seem to be a fiddle. You can easily google examples:

https://onlinetaxman.com/us-llc-tax-haven-for-foreigners/

" A US LLC opened by a non-US citizen or nonresident can arguably allow for earnings that are not taxed in the US. Effectively, foreigners are only subject to US tax if they are “engaged in a trade or business in the United States” (ETOB). If your business is not ETOB, even if it generates income in the US, the income is not taxed in the US."

This is an incredibly aggressive move. If it works, good luck. You would only be able to get real-world advice on this from someone who has done it many times, and that's not me.

Note too that the "other" country loves making an example of people about this sort of thing, typical article https://www.thetaxadviser.com/issues/2015/oct/problematic-use-of-transparent-us-llcs-by-foreign-taxpayers.html

Here's another article about exactly the question at hand

https://www.myusacorporation.com/articles/taxation-of-foreign-entrepreneurs#bm30

"I am a single owner of a U.S. LLC, non-U.S. person living abroad. My company provides remote services. Do I need to file tax return and pay income tax?

"A single member LLC [ ] would only pay tax based on the tax status of the owner. Since the owner is not physically present in the US and is providing services remotely there would be no income effectively connected to the US. That means the LLC would owe no US tax, except for the annual registration fee in the state of LLC registration, and there would be no US federal tax obligation (in other words there is no requirement to file income tax either). Keep in mind though - you might not technically be producing income in the U.S., but you [are almost certainly] liable to income tax on this income in your country.