I'd second Poppagator, except that I DO wait for inspiration. I find the songs I've written as an assignment tend to be "worthy" but not exciting. So I wait to be inspired, and it's not done me much harm.
There may be long gaps in my writing, but when I'm inspired a song takes me much less time to write. And I'm happier with it.

But no, not everyone needs to be a songwriter and not everyone should be. More than that, not every song you write has to be good, and not every song you write needs to be sung to an audience. Some songs are "practice" and some songs are "therapy" and there are a lot of writers who don't always know the difference between those songs and songs for public consumption! In addition to there being a lot of excellent songs out there, there's a huge amount of mediocre songs out there....it needs discernment and discrimination before deciding to put your songs in front of other people, and further discernment and discrimination to work out what they thought of them. In folk clubs, people always clap. Your family and friends will want to save your feelings. Keeps it all a question of your own judgment. Not easy.