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States that fail to certify their elections in time for the Electoral College do not participate because they failed to appoint Electors.

Again, if they fail to certify the electors then the process simply moves to the House and Senate with each picking the POTUS and VPOTUS respectively.
Each State will get to participate (just not in the EC aspect of it any longer) when the matter goes to the House and Senate. (see below)

...I cited the parts of the Constitution that shows the Electoral College requires a majority of appointed Electors, not a majority of *maximum possible* Electors.
You cited this...
The 12th amendment says "The person having the greatest Number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed..."
The "appointing" is done by Title 3 § 3. You're mistaking what it is saying.
Here is the rest of that which you didn't post...

The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; (that is where you cut it off, but it continues) and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice.
AGAIN...the number appointed is, by law, 538. The *maximum possible* electors is the same 538. (See Title 3 § 3)
I don't see anything anywhere stating that the number of electoral votes needed to win can be changed to 260. There is a set formula (see link below to USC 3) for determining the number of electors and the number of votes from the electors needed to win. Show me something...anything...that supports your view that the number of votes needed to win drops to 260.

You might also want to look at USC 3 which governs Presidential elections.
TITLE 3 — THE PRESIDENT § 3 - 5 are of major relevance. It isn't just the Constitution that is involved.

§ 4. Vacancies in electoral college
Each State may, by law, provide for the filling of any vacancies which may occur in its college of electors when such college meets to give its electoral vote.

It doesn't say the State has to fill any vacancies, it says it may.

§ 5. Determination of controversy as to appointment of electors
If any State shall have provided, by laws en-acted prior to the day fixed for the appointment of the electors, for its final determination of any controversy or contest concerning the appointment of all or any of the electors of such State, by judicial or other methods or procedures, and such determination shall have been made at least six days before the time fixed for the meeting of the electors, such determination made pursuant to such law so existing on said day, and made at least six days prior to said time of meeting of the electors, shall be conclusive, and shall govern in the counting of the electoral votes as provided in the Constitution, and as hereinafter regulated, so far as the ascertainment of the electors appointed by such State is concerned.
Pretty interesting info.

So here is what you didn't cite from Article II of the Constitution...
Article II Clause III (snip)

The Person having the greatest Number of Votes shall be the President, if such Number be a Majority of the whole Number of Electors appointed; and if there be more than one who have such Majority, and have an equal Number of Votes, then the House of Representatives shall immediately chuse by Ballot one of them for President; and if no Person have a Majority, then from the five highest on the List the said House shall in like manner chuse the President. But in chusing the President, the Votes shall be taken by States, the Representation from each State having one Vote; A quorum for this Purpose shall consist of a Member or Members from two thirds of the States, and a Majority of all the States shall be necessary to a Choice. In every Case, after the Choice of the President, the Person having the greatest Number of Votes of the Electors shall be the Vice President. The number "appointed" is 538 (See Title 3 § 3).

If you don't have enough (for whatever reason) of "a Majority" of those 538 votes (which would be the 270 required by law), the House and Senate decide the election.