6 months Visa Waiver for the UK?

I'll admit that I don't know everything but, as an experienced traveler myself with over a year of cumulative travel in Britain, I've never heard of this. Trying to see the logic in it but am at a loss.
Looking forward to reading more responses here on this topic.

Yes. When you enter Ireland you get 3 months. When you go from Ireland to the UK there is no border check so you are still traveling on that 3 month visa.

Remember visa free just means you don't have to apply for one ahead of time. The stamp you get at the airport is your visa.

So even if you are simply changing planes to the UK you have to transit through immigration control and get an Irish 3 month stamp?

What if I transit though Reykjavik and arrive in the UK? Is that three months?

Ireland is three months. There is a common border between Ireland and the UK.

If you go via Reykjavik, you will not go through immigration until you arrive in the U.K. (six months). You just transfer planes in Reykjavik, no immigration.

Reykjavik is in Iceland, not Ireland.

When I fly to Brussels through Reykjavik, I go through immigration and then exist Brussels airport without an immigration check. I thought the Common Travel area is only for Irish and UK passport holders.

The UK applies the CTA to all, so there are no regular immigration process arriving from the Republic of Ireland whether by land, sea or air.

The cases to be particularly careful of is if the purpose for coming to the UK requires a visa - as you can't get this done if arriving from Ireland.

But this may change if there is a hard border after brexit.

Actually anything, everything, and or nothing may change.

When I fly to Brussels through Reykjavik, I go through immigration and then exist Brussels airport without an immigration check. I thought the Common Travel area is only for Irish and UK passport holders.

Both Iceland and Belgium are in the Schengen Area. In that case you are entering the Schengen Area in Reykjavik, so no passport control in Brusels.
The UK and Ireland are not in Schengen, they are in their own "Common Travel Area", which is why there is no passport control on the Ireland/Northern Ireland border.
If you travel XXX -> Iceland --> UK you stay "airside" in Reykjavik (no passport control), but go through UK passport control when you land in the UK.

The situation being discussed is XXX --> Ireland --> UK. The UK allows 6 months for US citizens, but Ireland only allows 3 months. But they are AFAIK independent, so you could in theory do 3 months in Ireland, followed by 6 in the UK.

The Brexit problem is as follows: Both the UK and Ireland are in the EU, so no customs between them. And, with no passport check either (as above), there are no checks at the border, no need to stop.
The Good Friday Agreement, which ended the Troubles, says there can be no border checks. No problem, the EU + CTA does that already. But, if the UK exits the EU, that means customs between the UK and the rest of the EU (including Ireland), but on the land border that is not allowed, as per Good Friday Agreement.
If you can think of a solution, please send to Mrs T May, 10 Downing Street, London.

To answer the OP's question: "So even if you are simply changing planes to the UK you have to transit through immigration control and get an Irish 3 month stamp?"

I don't think so. If you change planes in Ireland and don't leave the airport, you don't clear immigration control until your final destination. For example, we landed in Dublin and had a layover of about 6 hours during which we stayed within the airport following the signs for passengers in transit. We then flew to Paris. We went through immigration passport control in Paris.

I may be wrong, but that is my experience and my understanding.

Procedures for Dublin to Paris (Schengen) are not the same as Dublin to UK (CTA).

Not sure what’s behind the OP’s question but yes, enter at Ireland you get 90 days, enter at the UK - up to 6 months.

This is the relevant UK law (The Immigration (Control of Entry through Republic of Ireland) Order 1972)-

4.—(1) Subject to paragraph (2), this Article applies to any person who is not partial and is not a citizen of the Republic of Ireland and who enters the United Kingdom on a local journey from the Republic of Ireland after having entered that Republic—
(a) on coming from a place outside the common travel area; or
(b) after leaving the United Kingdom whilst having a limited leave to enter or remain there which has since expired.
(2) This Article shall not apply to any person who requires leave to enter the United Kingdom by virtue of Article 3 or section 9(4) of the Act.
(3) A person to whom this Article applies by virtue only of paragraph (1)(a) shall, unless he is a visa national who has a visa containing the words “short visit”, be subject to the restriction and to the condition set out in paragraph (4).
(4) The restriction and the condition referred to in paragraph (3) are—
(a) the period for which he may remain in the United Kingdom shall not be more than three months from the date on which he entered the United Kingdom

Thanks, Marco, for the citation. Very helpful to read although I wonder what some of the language means like "local journey". My guess is that the intent of this law is to protect Northern Ireland and the UK from terrorist attacks at the height of the "Troubles". So
I am not sure why these rules are relevant today and why it should apply to Americans looking to save a few dollars by taking indirect flights to the UK via Ireland.

But regardless, it looks like the answer to the question I posed may be as follows:

If you are flying to the UK via a connecting flight in Ireland AND you pass through passport control at the Irish airport AND you DO NOT go through passport control at the UK airport, you will enter the UK with only an Irish passport stamp which only gives you permission to visit the UK for 3 months rather than the 6 months permitted under the visa waiver program between the UK and the US.

How can one avoid the 3 month limitation? Certainly one way is to take a direct flight from the States to the UK. And if flying to the UK via Ireland, check with the airline whether you will need to pass through passport control in Ireland even though you are simply getting on a connecting flight.

I also assume that flying indirect to the UK via any country other than Ireland will allow you to get your passport stamped for the UK.

Can anyone provide any corrections, further clarifications or additional details?

The parent legislation is the Immigration Act 1971. This allows the Secretary of State to change the terms of travel within the CTA relating to the UK by 'Statutory Instrument', ie not requiring a change to the actual Act.

The wording is actually more involved now as it has been amended a few times to add extra terms for visa nationals and to refer to EEA nationals, but the period remains unaltered for those outside these..