Could it be justified by the president's actions/inactions under the circumstances?
Right now? Heck no. The President is filing lawsuits to challenge the result, as well as making speeches about how the election was "stolen." The General Services Administration has also refused to cooperate with the Biden transition team, apparently on the grounds that the outcome of the election is still being litigated. All of that is highly irregular, but it is not extra-legal. Compare and contrast the 2000 election.
Section 4 of the 25th amendment was meant to cover situations where (for example) the President declares war on France because they are conspiring with the lizard people from Mars. We're nowhere near that threshold.
Could it be invoked quickly enough to prevent an orderly transition of power from going off the rails?
Yes. The 25th amendment takes effect immediately, and its wording implies that the President can only regain his powers after going through the whole process. This can take up to 25 calendar days in total (the VP must renew his claim of Presidential incapacity within 4 days after President challenges it, then Congress must vote in favor of the VP by a 2/3 majority within 21 days; if Congress does not so vote, there is no "fast-track" process to return the powers and duties early).
Would it depend on the co-operation of key individuals still loyal to the current president?
Probably. It requires Mike Pence plus a majority of the Cabinet. Most of those individuals are going to be loyal to President Trump.
Would it make any difference anyway? Would it only shift powers from the president to someone else still loyal to him and willing to do his bidding?
It would transfer power to Mike Pence, who is currently the Vice President. Trump selected Pence as his running mate in 2016, but Trump does not have the ability to fire or otherwise remove Pence. In practice, Pence has been quietly loyal to the President, but has avoided specifically repeating some of his more outlandish claims.