In addition to the chords listed in the OP, you should be able to play other common open chords, like dm, A7, E7, em7, B7, am7, D7, G7. Flipping through my guitar notebook, here are some songs with just a few chords and no barring. I’ve listed all the chords used in the song, but not necessarily in the right order.
Bob Dylan - “The Mighty Quinn” (G, D, C)
Bob Dylan - “I Shall Be Released” (C, dm, em)
Bruce Springsteen - “Atlantic City” (em, G, C, D)
Dolly Parton - “Jolene” (A, C, G)
“Long Black Veil” (G, D, C)
Oasis - “Wonderwall” (em, G, D, A, C, A7)
Oasis - “Live Forever” (G, D, am7, C, em)
Simple Minds - “Don’t You Forget About Me” (E, D, A, C, G)
Love - “A House Is Not A Motel” (em, G, am, C, D)
The Decemberists - “The Crane Wife 3” (D, A, G)
Joan Baez - “Diamonds and Rust” (em, C, G, D)
Joan Baez - “Love is Just a Four-Letter Word” (G, D, em, am, D7)
The Rolling Stones - “Ruby Tuesday” (em, D, C, D7, G, A7)
The Rolling Stones - “As Tears Go By” (G, A, C, D, em, D7)
Van Morrison - “Wonderful Remark” (G, D, C, em)
Van Morrison - “And It Stoned Me” (G, D, C, am, em)
Cat Power - “Metal Heart” (am, C, G, D)
Editors - “An End Has A Start” (am, em, C, D,G)
Flying Burrito Brothers - “Older Guys” (G, D, em, C)
“Will The Circle Be Unbroken?” (E, A, B7)
Steve Harley & Cockney Rebel - “Make Me Smile” (C, G, D, am)
Tons more open up if you can add just F and bm. Once you learn barring, then you can play almost anything. Really, the only way you’ll be able to play barre chords is if you make yourself play them. It is discouraging at first when you get a loud buzz or a muffled sound, but soon your hand will figure out the right position and develop the strength needed to play it right. It may be less frustrating to start with a song that uses all open chords except one (The Band’s “Up On Cripple Creek” has a f#m in the chorus; “Victoria” by The Kinks has a bm). When it’s just one chord, it makes less of an impact in the song if you don’t play the chord perfectly at first. Hopefully, you’ll be confident enough in your open chords to motivate yourself to get that one barre chord just right so you can play the whole song start to end.