Continuing with John Smith, Joiner, here he is in 1841: HO107/511 9/38 pg 27 at Wavertree Green, Childwall, Wavertree Lancs (West Derby District):
Thomas SMITH 55 Gardener Y
Jane SMITH 50 Y
John SMITH 20 J. Joiner Y
Agnes SMITH 11 Y

He's probably not married yet and these may be his parents and sister.

There are just a handful of John Smiths marrying a Mary in the West Derby District in the 1841-1845 range, but one of them is significant.
From BMD Index: Marr's Dec 1841 W.Derby 20 698 there is a John SMITH and a Mary NIGHTINGALE. Your Elizabeth Scarffe nee Smith had a daughter Florence Nightingale Scarffe. On first blush I figured she named her after the famous nurse, but more likely Nightingale is a family name. It's not on the Scarffe side, so it must be on Elizabeth's side. This is getting better all the time!

I think this is Mary Nightingale in 1841, at Hills Place, Childwall, Wavertree HO107/511 9/28 pg 7
Mary NIGHTINGALE 25 Y
John HITCHMOUGH (?) 20 Jo Joiner Y
Elizabeth 20 Y
Alice RICHARDSON 35 Female Servant Y

I found John and Mary Smith in 1881 at 10 Pennington Cotts, Wavertree. I only looked at the extraction and not the original:
John SMITH Head Mar 62 Carpenter b Wavertree
Mary SMITH Wife Mar 67 b Wavertree
John BURNS, Lodger 23 Gardener (Dom) b Wavertree

They don't seem to be in 1891, so I checked the Death index and there are one or two John Smiths West Derby who are the right ages between 1881 and 1891. Same thing for his wife Mary. It wouldn't be difficult to find the right ones, since it likely will be at Wavertree.

Now, we really must try to find Elizabeth Smith born ca 1845 in the 1861, 1871 and 1881 census's, and see if she will match into this family.

This proves it is possible to find and track a John Smith, which must be the most common name of them all.