Regarding Salah, it should be performed in the strict manner ordained by the Prophet Muhammad(p.b.u.h) and only the prescribed verses should be recited at the prescribed times.
The Prophet (ﷺ) said, "Pray as you have seen me praying."
Sahih Bukhari, Book 10, Hadith 28
Any Quranic verses (and only Quranic verses) may be recited after the Fatiah and before the Rukuh(bowing down). There is no evidence for the Prophet reciting Hadith within the Salah so we don't do that. In matters of obligatory religious rituals and practices anything that the Prophet didn't do is considered an innovation(Bid'ah) and is sinful.
The Messenger of Allah (ﷺ) said, “He who innovates something in this
matter of ours (i.e., Islam) that is not of it will have it rejected
(by Allah).” [Bukhari & Muslim] In another version in Muslim it reads:
“He who does an act which we have not commanded, will have it rejected
(by Allah).” Fourty Hadith Nawawi 5
Furthermore there is no evidence that reciting Hadith has any virtue in itself. Hadith are meant to only be studied for the teachings in them and the words themselves do not have any virtue attached to them.
On the other hand reciting Quranic verses in Prayer is proven from several sources, including the Quran itself:
Quran 73:20 Indeed, your Lord knows, [O Muhammad], that you stand [in prayer] almost two thirds of the night or half of it or a third of
it, and [so do] a group of those with you. And Allah determines [the
extent of] the night and the day. He has known that you [Muslims] will
not be able to do it and has turned to you in forgiveness, so recite
what is easy [for you] of the Qur'an. He has known that there will be
among you those who are ill and others traveling throughout the land
seeking [something] of the bounty of Allah and others fighting for the
cause of Allah. So recite what is easy from it and establish prayer
and give zakah and loan Allah a goodly loan. And whatever good you put
forward for yourselves - you will find it with Allah. It is better and
greater in reward. And seek forgiveness of Allah. Indeed, Allah is
Forgiving and Merciful.
And in Hadith:
Narrated Abu Huraira: The Qur'an is recited in every prayer and in those prayers in which
Allah's Messenger (ﷺ) recited aloud for us, we recite aloud in the
same prayers for you; and the prayers in which the Prophet (ﷺ) recited
quietly, we recite quietly. If you recite "Al-Fatiha" only it is
sufficient but if you recite something else in addition, it is better.
Sahih Bukhari, Book 10, Hadith 166
Narrated Ibn Abbas: (My mother) Umu-l-Fadl heard me reciting "Wal
Mursalati `Urfan" (77) and said, "O my son! By Allah, your recitation
made me remember that it was the last Sura I heard from Allah's
Messenger (ﷺ). He recited it in the Maghrib prayer. Sahih Bukhari
Book 10, Hadith 157
