According to R. Meir Wunder (book review of Or Hagalil – in Hatzofe 10 Av 5736 – cited in TMOAG page 688) the "custom" of traveling to Amuka looking for a Zivug is no older than 1953 – when a tour company responded to the pleas of a spinster – to find a special place to pray for a match.
The claim that Rav Yonatan ben Uziel is buried in Amuka can be found in a document 1000 years after his death – the veracity of this tradition has been challenged. This document was only revealed in the 20th century as it had been in the Geniza in Cairo
The claim that Rav Yonatan ben Uziel was never married has no basis
The claim that he told people to pray at his grave apparently has no basis
I think the way it works is a lot of single people go there…
See The Making of a Gadol Rabbi Nathan Kamenetsky volume one page 688-689 and footnotes