The hadith must be regarded in the light of the initial hadith of the chapter
Well you claim you are not cherry picking, but the first hadith in this chapter in Sahih Muslim says:
When the slave-woman of any of you commits adultery and this (offence of hers) becomes clear, she should be flogged (as the prescribed) punishment, but hurl no reproach at her. If she commits adultery again, she should (again be punished) by flogging, but hurl no reproach upon her. If she commits fornication for the third time and it becomes clear, then he should sell her, even if only for a rope of hair
Which indicates that the pre-requirements for a punishment for committing zina are fulfilled.
Imam an-Nawawi explains in his commentary on Sahih Muslim -see here- saying:
Here and in the follwoing I'm translating from Arabic language, as these translations are of my own take them carefully!.
ومعنى تبين زناها : تحققه إما بالبينة وإما برؤية أو علم عند من يجوز القضاء بالعلم في الحدود
"and this offence of hers becomes clear" means: it became clear by evidence either by eye witness or with whom it is permissible to judge with knowledge within the limits.
Al-Qurtoby explains it as follows in his commentary -see here- on Sahih Muslim:
وتبين زنى الأمة يكون بالإقرار وبالحبل، وبصحة الشهادة عند الإمام.
The zina of the slave girl is made clear by confession and by conception, and by the validity of the testimony according to the imam.
So witnesses are among the necessary pre-requirements for condemnation!
There's one difference of opinion among the scholars, it is whether the slave owner can punish once he has the "evidence" or the imam is necessary to give a final judgement to apply the punishment.
Witnesses are already requested in the qur'an
And Allah the Almighty ordered:
Those who commit unlawful sexual intercourse of your women - bring against them four [witnesses] from among you. ... (4:15)
And those who accuse chaste women and then do not produce four witnesses - lash them with eighty lashes and do not accept from them testimony ever after. ... (24:4)
Why did they [who slandered] not produce for it four witnesses? ... (24:13)
It is agreed upon that four (free, Muslim) male witnesses are necessary for a condemnation for zina, based on this verse. However the ruling on slaves has a difference of opinion when it comes to the punishment based on whether the slave was married or not and who was involved in this act of zina. This is also based on the interpretation of "الإحصان" in the verse from surat an-Nisaa'.