The hadith of ʿĀʾishah is a ḥasan ṣaḥīḥ hadith.
And the practice upon this is [followed] by some of the people of knowledge.
And it was narrated from Ibn ʿUmar that the Prophet ﷺ performed ḥajj alone (ifrād), and Abū Bakr, ʿUmar, and ʿUthmān performed ḥajj alone.

834 - Qutaybah narrated to us that, he said: ʿAbd Allāh ibn Nāfiʿ al-Ṣāʾigh narrated to us, from ʿUbayd Allāh ibn ʿUmar, from Nāfiʿ, from Ibn ʿUmar, with this (1).
And al-Thawrī said: If you perform ḥajj alone (ifrād), that is good; if you perform qirān, that is good; and if you perform tamattuʿ, that is good.
And al-Shāfiʿī said the like of that, and he said: What is most beloved to us is ifrād, then tamattuʿ, then qirān.

11 - Chapter: What has been narrated concerning combining ḥajj and ʿumrah
835 - Qutaybah narrated to us, he said: Ḥammād ibn Zayd narrated to us, from Ḥumayd, from Anas, who said: I heard the Prophet ﷺ say: "Labbayka bi-ʿumratin wa-ḥajjah" (Here I am, at Your service, for ʿumrah and ḥajj) (2).
And in this chapter [there is a narration] from ʿUmar and ʿImrān ibn Ḥuṣayn.

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(1) Recorded by Aḥmad (5719) and Muslim (1231) (184) without his saying: "And Abū Bakr, ʿUmar, and ʿUthmān performed ḥajj alone," and it was recorded in full by al-Dāraquṭnī in his "Sunan" 2/239.
(2) A ṣaḥīḥ hadith, recorded by Muslim (2151), Abū Dāwūd (1795), al-Nasāʾī 5/105, and Ibn Mājah (2969), and it is in the "Musnad" (11958).