There’s a song I finally remembered to share with Mr. All-I-Wanna-Be-Is-A-Dad, aka my Middle Bro, and he says he bawled like a baby reading the lyrics, according to Google it’s sung by George Strait: “Love without end, amen”. Being country, it’s got relatively simple lyrics (I can barely understand some of the lyrics that have been quoted, guys, this is for a class).
And we have the trifecta! 
This is about a stepfather, not a biological father, but it’s a great song anyway, and seems to fit your theme: “He Didn’t Have to Be” by Brad Paisley
Country music has a metric shit-ton of Daddy songs. Here’s a couple rather glurgy ones that I still like:
And a couple of heartbreakers from Tim McGraw:
Walk Like A Man - this is not a cover of earlier songs by the same title, BTW. Very sad stuff:
His old man woke him up to help him find the car
Take the bus uptown outside some rundown bar
And swore that he was quittin
And then he’d start to cry
And though the boy knew the truth
Sometimes he’d still believe that lie.
And, from his new album: You Had To Be There about a young man meeting his father for the first time when he (the son) is in prison.
I’d also recommend either version of Forever Young. Me, I like the Rod Stewart version best. My son and husband prefer Bob Dylan.
Sorry - I thought that cutting out 80% of the song would satisfy those requirements. The famous bit’s cut out now. 
And it never occurred to me about Kipling and Thomas; I’d be amazed if Kipling was still in copyright but I guess Thomas still is.
I just checked - Kipling’s works are definitely out of copyright as he died in 1936. Thomas’s might be, but I’m not sure. I understand moderators not wanting to check each individual poem, though.
Life + 70 years for UK copyright, no? Thomas died in 1953 I believe, so Kipling just makes it and Thomas doesn’t.
It’s not quite as simple as that, hence Thomas’s works possibly being in copyright. They probably are under the 70 year rule too, but it’s not certain; the mods are right to err on the side of caution with his stuff. Kipling’s safe, though. Of course, when you hear old poems being used in ads it generally means they’ve just come out of copyright.
The nauseating “Watching Scotty Grow” by Bobby Goldsboro. It was put to good use on a Simpsons episode where Homer and Bart enter a fighting robot competition.
Bnl’s When You Dream
I had never thought of “If” as being from a father to a son; I always took the “my son” at the end to be just a diminutive applied to a child by any wise adult.
Have I got it wrong all these years?
It can be interpreted either way. I prefer the literal father/son interpretation because the advice reminds me of Polonius and Laertes and also because Kipling was posh; I can’t imagine a posh person using ‘my son’ as a term of affection the way commoners do.
Yet another country number:
You didn’t mention how old your ESL students are, but if they’re in their teens, they could get a chuckle from it.
The boys and girls still sit at separate tables. The fear of “cooties” runs pretty deep in this land. They range from about 11-15.
Song for my Father - Angels of Light (Michael Gyra)
When the warm wind blows through the corridors and tiled walls,
There you are, the mad guardian of your empty room.
Your memories were tortured by the scent of ocean through the halls
and where was I, the one that you loved best ?
Now show me how to hold the knife that cuts the page,
to crucify the weakness and the guilt I use for faking strength.
Now I remember, I crawled naked across the broken stage,
and there was I, the child that brought us shame.
So walk across the blood red ocean,
Stand above the broken glass and concrete towers.
In here I sleep, stoned guardian of my empty dream.
Now I’ve become the child that neither of us knew.
Thank God you never knew the things that I have done.
Thank God you never saw the person I’ve come.
“Walk Like a Man” by Springsteen has been mentioned twice, but you could write a thesis based on all of the songs he’s written about his relationship with his father. Two great ones:
“Independence Day”
“My Father’s House”
Peter Gabriel’s Father and Son
The topic is also explored tangentially in “Used Cars”
John Lennon’s Beautiful Boy is a great song that he wrote for his son.