Well the current Lord Leigh is a descendant of the Lord Mayor and has a middle name Piers, and so did his father. But I can't see where else any descendant of the Lord Mayor was called Piers, so it doesn't look like the Agincourt thing was a big family tradition.
Burke's Commoners (1836) has a long article on Chandos Leigh, before he was a Baron. He was a great-grandson of Theophilus I, so a 2nd cousin of Jane Austen.
https://archive.org/stream/genealogicalhera03burk#page/223/
The article runs through all the main lines of descent from the Lord Mayor. Mentions Agincourt in a footnote.
Part of this article is what is recycled in Extinct Baronets (1841). The Stoneleigh baronets aren't covered in earlier baronetage books, because they became barons, so they were left to the peerage books. And they disappeared from the peerage books when they became extinct in 1786.
It's quite likely that Burke's got it from an old Collins Peerage, but only bits of those seem to be online. Collins mostly got his info from the families, unless they were unresponsive, in which case he had to resort to research, but not in primary sources. His books were pro-aristo propaganda, so he wasn't the man to disagree with anything Their Lordships told him.
A long pedigree compiled by somebody has found its way into this so-called "Visitation of Cheshire" manuscript
https://archive.org/details/visitationofches00glov/page/154
according to which the Lord Mayor was the great-grandson of John Legh of the Ridge, who married the heiress Alice Alcock.
This John is then hooked up as the son of Piers I by a 2nd wife, Cicely del Hagh. But that can't be right - Piers I's first wife Margaret Danyers outlived him.
The best account is this history of Prestbury by Frank Renaud
https://archive.org/stream/contributionsto00renagoog#page/n167/
According to which, there was a brass in Macclesfield church which said Piers II died with Henry V in Paris, and an obscure alternative version that said he was killed at Agincourt. People have tried to have it both ways by saying he died of his wounds. But it's quite possible he died of dysentery, like the king and hundreds of others. Bravery was drinking the water.
Page 154 goes through the theories about the origins of John Legh of the Ridge
https://archive.org/stream/contributionsto00renagoog#page/n192/
Apparently it was Dugdale who said John was the grandson of the first Piers Legh of Lyme, which could make him the son of Piers II and support Burke.
Renaud is adamant that John was the brother of Piers I. He proves that Maud Norley had a younger son called John, though I don't see where he proves this is the same man who was escheator of Cheshire in 1453, when he would have been at least 83, as Maud Norley's husband Robert Legh of Adlington died in 1370.
Renaud then contradicts the Vis Cheshire chart again by saying that the Leighs of Stoneleigh (ie the Lord Mayor) descend from Roger of the Ridge d 1506, son of Roger d 1448, son of John, not from his cousin Roger of Wellington, son of Richard of Rushall, son of John.
Which gives us a choice of two Rogers for the Lord Mayor's father.
For what it's worth, this "1568 Visitation" pedigree says the Lord Mayor's mother was a Trafford
https://archive.org/stream/visitationoflond00cook#page/10/
But it also mentions 1610. The explanation is that this is a copy made by Nicholas Charles (d 1613) with his own additions, in the same handwriting, so we can't tell how much is original evidence.
