Translation:I ran home yesterday and I ate my dinner.
If I spoke this same sentence but without the "mé" after "d'ith", would it still mean the same thing?
Yes. You see it all the time in writing. From the book I'm currently reading:
Shuigh Mortimer síos arís agus thosaigh ag scríobh go sciobtha.
And it's not just a new feature from English influence; Padraic Ó Conaire, perhaps the first writer of modernist Irish fiction, used it frequently too in his short stories from the early 20th century.
Perhaps it’s a (now) old feature from English influence? Wasn’t Ó Conaire an anglophone who’d learned Irish at school?
From what I remember, Ó Conaire was born in Galway, but his family was from Rosmuc, and he lived there for a while with his uncle. So it's likely he was probably bilingual. Meaning it could be from English influence. I haven't really read many older works recently to verify this.
This really isn't a dialect switch - it's a normal effect of running the long é up against the a in abhaile. And even for novices, rith mé should be such an obvious combination that it shouldn't cause too much confusion.
(And to be perfectly honest, I'm not really sure what's up with Sinéad's pronunciation of mé as "meh" on teanglann.ie).