The hadith even if it was qualified as da'if on sunnah.com has been quoted (at least partly) in several other hadith compilation such as a second narration in Jami' at-Tirmidhi, in Sunan ibn Majah, in al-Bayhaqi's as-Sunan al-Kubra (here in Arabic, where he points that the issue is Hamza ibn Maymoun who has been qualified as da'if by Ahmad ibn Hanbal and ibn Ma'yn and later scholars), in at-Tabarani's al-Mo'jam al-Kabir (see here in Arabic), in ad-Darimi's Sunan (see here in Arabic) and in those of ad-Daraqotni (see here in Arabic) all of these narrated it via more or less the same chain. Better narrations or partial narrations of this hadith appear in the Mosnaf's of ibn abi Shaybah (here in Arabic) and 'Abdurrazaq and in abu 'Obayd's an-Nasikh.
(Sources this thread and this article both in Arabic)
The verse our prophet () quoted in some of these narrations is:
Righteousness is not that you turn your faces toward the east or the west, but [true] righteousness is [in] one who believes in Allah , the Last Day, the angels, the Book, and the prophets and gives wealth, in spite of love for it, to relatives, orphans, the needy, the traveler, those who ask [for help], and for freeing slaves; [and who] establishes prayer and gives zakah; [those who] fulfill their promise when they promise; and [those who] are patient in poverty and hardship and during battle. Those are the ones who have been true, and it is those who are the righteous. (2:177)
This quote includes more than zakah: It includes giving from ones wealth in spite of loving what one is giving, it also make clear that if one is able to one should use the wealth which is aside zakah for needy people and for doing good deeds.
Al-Mubarakpuri in his Tohfat al-Ahwadhi تحفة الأحوذي (see here in Arabic) considers these kinds of "gifts" that are mentioned in the verse above as obligatory for a wealthy person.
This is mentioned elsewhere in the qur'an:
And those within whose wealth is a known right (70:24)
and goes ahead with ahadith such as:
I reached him (the Prophet (ﷺ) ) while in the shade of the Ka`ba; he was saying, "They are the losers, by the Lord of the Ka'ba! They are the losers, by the Lord of the Ka'ba!" I said (to myself ), "What is wrong with me? Is anything improper detected in me? What is wrong with me? Then I sat beside him and he kept on saying his statement. I could not remain quiet, and Allah knows in what sorrowful state I was at that time. So I said, ' Who are they (the losers)? Let My father and mother be sacrificed for you, O Allah's Messenger (ﷺ)!" He said, "They are the wealthy people, except the one who does like this and like this and like this (i.e., spends of his wealth in Allah's Cause).
(Sahih al-Bukhari and elsewhere like in Sahih Muslim)
Ibn 'Abbas told Ibn az-Zubayr, "I heard the Prophet, may Allah bless him and grant him peace, say, 'A man is not a believer who fills his stomach while his neighbour is hungry.'" (In al-Bukhari's al-Adab al-Mufrad and also compiled by at-Tabarani)
So a good wealthy Muslim should consider sharing from his wealth. He shouldn't say deny those who ask him (for food, beverage, help, money etc.) if he is able to give it. Zakat is only the due part which helps us overcome our natural born meanness some scholars say.