All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu 'alayhi wa sallam, is His slave and Messenger.
If a person performs only the pillars and obligatory acts of the prayer, without performing the acts of Sunnah and recommended acts, his prayer is valid. However, abandoning the recommended acts of the prayer altogether is disliked. This is because the Muslim is required to follow the example of the Prophet regarding all acts of the prayer that were authentically reported in the Sunnah. He made his prayer our reference regarding how to pray, saying: “Pray as you saw me pray.” [Al-Bukhari]
Some scholars stated the gravity of habitually abandoning the acts of the Sunnah. Therefore, a Muslim should keenly follow every act of the Sunnah which is authentically attributed to the Prophet especially in the prayer, it being the main cornerstone of the religion. He should beware of going against the Sunnah, as Allah, The Exalted, Says (what means):
{There has certainly been for you in the Messenger of Allah an excellent pattern for anyone whose hope is in Allah and the Last Day and [who] remembers Allah often.} [Quran 33: 21]
{And whatever the Messenger has given you – take; and what he has forbidden you – refrain from. And fear Allah; indeed, Allah is severe in penalty.} [Quran 59: 7]
The Hadeeth to which you referred was cited in Sunan At-Tirmithi and Sunan Ibn Maajah, and it reads: The Prophet said, "The first thing (deed) a person will be held accountable for on the Day of Judgment will be his prayers. If they are good (properly performed), he will have prospered and succeeded, and if they are defective, he will have failed and lost. If there is something missing in his obligatory prayers, the Lord Almighty will Say, “See if My slave has (performed) any voluntary prayers with which to complete the deficiency in his obligatory prayers.” Then, the rest of his deeds will be judged in the same manner.’”
Therefore, a Muslim should establish his prayers observing all its pillars, obligatory acts, and recommended acts in order for it to be perfect and complete. Although it is true that the non-performance of these recommended acts does not invalidate the prayer and the person is not considered sinful by leaving them out, he is nonetheless missing abundant good, and it is feared that he would eventually end up abandoning the pillars and obligatory acts of the prayer as well, Allah forbids.
Allah knows best.