Yes, those two popular stories should not be presented as authentic hadith. But their weakness proves a problem in some preaching, not a problem in the established historical character of the Prophet (PBUH).
Why the criticism has force
The strongest version of the criticism is fair: if a preacher says, "There is a hadith that an old woman threw garbage on the Prophet (PBUH) every day," or tells the baggage story as established seerah, he is making a claim that needs evidence. A warm lesson is not enough. In Islam, reports are weighed by chains, texts, and the judgment of hadith scholars.
As far as the well-known garbage story and the old woman fleeing Mecca story are concerned, they are not found with authentic chains in the reliable hadith collections. Many teachers repeat them because they heard them from others, not because they checked them. That should stop. Love for the Prophet (PBUH) is not served by attributing unsourced tales to him.
What this actually establishes
It establishes that Muslims must be more careful with popular stories. It may also expose lazy da'wah: using sentimental anecdotes when authentic material is already more than enough.
But it does not establish that the noble character of Prophet Muhammad (PBUH) is invented. That is a much larger claim, and it fails. The Quran itself describes his character in clear terms:
And indeed, you are upon a tremendous character.
Quran 68:4
And We have not sent you except as a mercy to the worlds.
Quran 21:107
Then the authentic Sunnah gives many concrete examples, preserved through the very critical method that rejects the weak stories.
The authentic record is stronger than the legend
Aisha, who knew his private life better than almost anyone, described his temperament:
The Prophet (PBUH) was never given a choice between two matters except that he chose the easier of them, so long as it was not sinful. If it was sinful, he was the farthest of people from it. And he never took revenge for himself, but if the limits of Allah were violated, he would take action for Allah.
Sahih al-Bukhari and Sahih Muslim
After Ta'if, when he had been mocked, expelled, and injured, the angel offered to crush the people between the mountains. His answer was not revenge:
Rather, I hope that Allah will bring forth from their descendants people who will worship Allah alone and not associate anything with Him.
Sahih al-Bukhari and Sahih Muslim
When a Bedouin urinated in the mosque, the Companions rushed to stop him. The Prophet (PBUH) restrained them, allowed the man to finish, and taught him gently, then ordered water to be poured over the place.
Leave him, and pour a bucket of water over his urine, for you have been sent to make things easy and not to make things difficult.
Sahih al-Bukhari
He also visited a sick Jewish boy who had served him, sat by his head, and invited him to Islam with concern for his salvation.
The Prophet (PBUH) came out saying, "Praise be to Allah who saved him from the Fire."
Sahih al-Bukhari
A fair way to speak about it
So the honest answer is simple: do not use the garbage story or the baggage story as proof. If someone says, "They are famous, but I do not know an authentic source for them," that is better than beautifying religion with unchecked tales.
Islam does not need fabricated sweetness. The verified life of the Prophet (PBUH) already contains mercy, patience, courage, restraint, and truth. Weak stories should be retired, and the authentic record should be taught with confidence.