The value of apologizing is a function of the type of people you are dealing with. If you apologize to people who respect the truth, an apology is considered strength, since it shows that you are objective enough to recognize and respect the truth. If you apologize to people who prefer to lie and deceive, this is a sign of weakness, and your apology will be spun with another lie.
The left wing media is 93% negative against Trump. There is no up side to an apology to such people, since they lack any sense of balanced objectivity to reality. An apology to an irrational person, is not a lesson in truth or humility, but can be seen as a new opportunity to spin, driven by the negative underlying emotions. You will get 93% negative reviews. Apologies work best with objective people who know right from wrong, and who know, that you now know, right from wrong.
When it comes to politics, opinions can be totally opposite; pro or anti-Trump. How do you apologize for believing in A, to an opponent who believes B? To apologize, you would have to deny what you believe is true, to help others reinforce, what you believe is false, so they can feel you found the truth. Apologies do not work in politics since truth is relative, and no lesson in truth is ever learned.
For example, if Trump apologized each time the left brings up mud; apologize to appease the leftist hypocritical bias, the left would arrange to have a daily mud bath for a Trump apology. Whether true or not, this would make Trump look like someone, anyone can emotional bully into a submissive role. This weakness then attracts bullies.
Because Trump does not apologize, he nips this bully lure in the bud. Now the easy scam, needs a more elaborate plan and much more use of time and money, yet it may still not get any long term traction. Apologies do not work with people whose heart is full of hostility and negativity, since they will use these emotions when reacting to the apology. Fear works better for them, since fear trumps hate and negativity. The fear (of God) is the beginning of wisdom and truth. You can apologize to people whose heart have love, since they will interpret the apology with love.