There have been many singers who did not write their own songs, over the last few decades. Some would be considered superstars or famous. There also have been songwriters who were not good singers. Some sang their own music they had written but they never became known as great singers.
Then there are singer/songwriters. They are not necessarily better or worse than “mere” singers or “mere” songwriters. They are, however, throwbacks to yesteryear, decades or centuries ago. Way back when, folk-ish singers had to write their own music and there were no computers, radios, or concerts within their reach from which they could learn or copy another’s music. They had an instrument, such as a guitar, lute, or other portable chord making thing and they played it as they sang.
To be clear, even though singing and songwriting are related, they are multiple skill sets. Most songwriters play keyboard and/or guitar well. They play well enough so as to be able to write chords and rhythm patterns which make stylistic sense. There are patterns on piano and there are strum or pick patterns on guitar. In addition, there may be fills and embellishments used by songwriters to enhance a song. Further steps would be orchestration and/or arranging which are still other skill sets.
The melody of the song and melody writing is another separate skill set. Writing melody and chords are sometimes learned or enhanced by having the knowledge of music theory. There have been many good or great songs written by people who only play by ear, know what they hear, know what they like and it doesn’t make the song suffer at all. What if… what if a person gets stuck in the process? A lack of knowledge can cause a person to get stuck because melody writing and harmonic technique will only be as good as experience plus knowledge in those areas.
Some songs have one chord or two and most have more, but there are tendencies of many progressions. A I-IV-V progression is found in thousands of songs but the interesting thing is none of the songs have the same melody lines at all. II-V7-I progressions are interesting and can completely change a song. So can II-V7-III-Vi. Then there are chords way beyond major and minor and these all can open up what feels like an entire universe of sound.
The more you know, the more you can do. How long does it take to learn chord formation? That would be a great first step. I once taught a non-musician all chords in every key in 3 hours. I could call out a chord and she would play it with little hesitation. Is it worth the time spent to be able to do this? What if it took 8 hours or a week? Is it worth spending the time to improve?