You also can’t have your Easter Sunday without your Nativity, but there’s no argument about singing Christmas carols!

Part of picking the songs is to reflect what’s being celebrated in front of you right now, not the themes that were celebrated that led up to where you happen to be, but were already celebrated elsewhere a few days ago.
Triduum is sort of like a liturgical entity that encompasses multiple physical days, which is part of why Holy Thursday has such a unique end to it-- it’s not an end, per se, but rather, more like going into recess until you pick up again a bit later. It has a nice flow that culminates in the Easter Vigil on Saturday evening, and by the time you get to Easter Sunday at Dawn, or some other daytime Sunday Mass, you’ve gotten past all of your Triduum emotion and energy, because you’re into the third or fourth Easter celebration at that point.
So referencing a Good Friday song at Easter Vigil is, perhaps, the liturgical equivalent of serving bagels and scrambled eggs at dinnertime. It’s unseasonal, if that makes sense, even if the song is a good song, and the food is good food. Referencing a Good Friday song on Easter Sunday is like having a Thanksgiving spread on the second Sunday of Advent… it’s still a good song, and it’s still good food, but it’s really unseasonal by that point.
–edit-- Although, I do admit, it would make sense to incorporate it if you were music minister at a Protestant church that doesn’t celebrate Holy Thursday or Good Friday as distinct feasts. If they wanted to smush the entire Triduum-- Holy Thursday, Good Friday, Holy Saturday, Easter Vigil, etc-- into one celebration, then yes, it would make sense. But hymns like “Lift High the Cross” would be more appropriate, because it’s about victory, rather than suffering. My parish is fond of “The Strife is O’er, the Battle Done”, “Jesus Christ is Risen Today”, “Crown Him With Many Crowns”. In contrast with more Good Friday-ish songs, like “Were You There”, “Old Rugged Cross”, “O Sacred Head Surrounded” and so on.