It's not that Morrissey's 'part of the song' wouldn't exist without Marr's part; it's that Morrissey's (Smiths) songs wouldn't exist without Marr's music. In that sense, Marr's music was of course essential - Morrissey never created a vocal melody for his words out of thin air, as far as I'm aware. Marr's music wasn't always essential to the lyrics, because sometimes Morrissey had written them already, but it's only Morrissey coined a vocal melody for the lyrics that the song came into being. Before the vocal melody and lyrics are created, there is no song, just anonymous music - Had Marr given his (untitled) backing track for 'This Charming Man', to choose an example at random, to any other song writer - let's say Matt Johnson of The The - then the song 'This Charming Man' would never have existed. Matt would have added different words and a different vocal melody to create whatever it was titled - 'Song X' let's say. Once that was done, strip away the backing music to leave only vocal melody (let's say, for example, somebody did an acappella version) and you'd still have 'Song X'.
Therefore 'This Charming Man' is Morrissey's song because its Morrissey's. But it owes its existence as much to Marr as it does Morrissey. For that to be the basis of the co-credit though (which it basically is), you might Morrissey's mum and dad as well, for giving birth to him, ditto Marr's parents etc etc.
The credits on Smiths records should have read as follows:
'Songs by Morrissey'; 'Music by Marr/Rourke'. And sometimes 'Music by Marr/Rourke/Porter' or whoever else chipped in on the demo before Morrissey added his words/vocal melody.
But of course, that was never going to happen. It's interesting though that the credit did change between first and second album, from 'Words by Morrissey, Music by Marr' to a more ambiguous 'Songs by Morrissey/Marr'. I think Morrissey felt that the 'Words' credit didn't reflect what he regarded as his true to the songs - which it didn't. But in fact he was still being done out of the fuller credit that he deserved -'Songs by Morrissey'.