Melodic considerations aside, I don't have a problem with that. I'm not a big fan of updating old texts to reflect modern sensitivities, but the issue with religion is that it is didactic, and "a wretch like me" is just an awful concept plus a worse teaching. I would never teach or imply that humans by their "fallen nature" are "wretches" or "depraved" or whatever. And it's a really terrible thing to teach a child, so since it in a didactic context I applaud these updates.
I think the issue here is that no one is forcing a church to sing a hymn with disagreeable theology. by changing the lyrics, one warps the original intent of the author and I feel that's disrespectful to the author and their work.
Take Amazing Grace. John Newton was a former slave trader. Regardless of one's views on depravity, it's fairly obvious that Newton thought
he was a wretch. Sanitizing that line cheapens his intent in writing the hymn. Again, this isn't a question of theology, but appropriate use of an author's work. There are plenty of acceptable hymns to use if one doesn't believe in original sin.
The only instance of this I've seen in my church was a Catholic hymn that contained a line about communion that was obviously Catholic. Some Protestant had changed the line to something completely unrelated that didn't work and it stuck out like a sore thumb. It cheapened the original intent of an author who fervently believed in that bread and wine are Christ's body and made an otherwise beautiful hymn sound shoddy. Rather than requiring an update, this hymn just wasn't appropriate in a Reformed setting, and should've been set aside in favour of more theologically acceptable hymns.