I'm not sure. All of my favorite artists write their own music, but I feel like even if they hadn't written any of it, I would still love the music just as much. Because the song would still be the same.

For example, I always praise and worship Tiziano's incredible poetry and songwriting, but on his latest album he covered a rap song, and to me it's up there with his best work.

The thing is that the vast majority of artists have no choice but to create their own material. It's only a select few popular stars who have the luxury of getting songs handed to them. So the probability that I like a singer-songwriter is automatically a lot higher, because I don't exclusively listen to top 40 radio to get my music fix.

It doesn't matter to me at all.

When i listen to a song, I can't tell just by listening, whether that particular artist wrote it themselves. All I can tell from listening to a song, is whether I like its' sound and the song resonates with me. So I don't think it's important at all when enjoying a song tbh. That said, I think it does indicate talent if a person can write a beautiful song like "Someone Like You" etc.
But on the other hand, it really doesn't mean anything that an artist writes their own songs, if they aren't that lyrically and sonically clever and enjoyable, and don't sell.