I think that trying to make a living as just a songwriter is going to be VERY difficult! In the very early days of records, the majority of artists did NOT write their own songs. But, that changed quite a while ago and these days most bands and artists do write their own songs. However, there are still plenty of artists looking for great songs to record for their products, but there are WAY too many songwriters out there trying to get their songs placed.
Your songs have to be exceptionally good if you are ever going to get any of them placed. Plus, unless you are working with a big publishing company that can get your song in the hands of big name artists and producers, then you pretty much have no chance at all.
At the minimum, I think you need to be able to play piano or guitar really well in order to write a good song because a great song needs more than just words. You need a great melody and chord progression to go along with it.
Also, if you want to be the best writer you can be, then you should be out there taking songwriting courses, and music composition and theory courses, and learning everything you can about music and songwriting in all styles. The more you know, the better chance you have of writing a great song.
You don't need to learn how to be a great recording engineer, or a great singer, or how to produce beats... While all those things may be helpful and would allow you to make demos of your songs for cheap, it isn't absolutely necessary. In fact, the more time you spend on trying to learn all those other things, the less time you will have to focus on becoming a better songwriter.
You should check out my article entitled "Can you really do it all yourself" on this web site.
Now, it used to be that if you had a really great song, then a simple piano or guitar and vocal demo was more than enough. Producers and Record Company people used to be able to recognize a great song from even a very simple and poorly recorded demo. However, these days with the proliferation of cheap recording gear, people are now used to hearing fully produced and professional sounding song demos. So, it's a tough choice as to which way to go. Some producers still like to hear very raw and simple demos since then they aren't distracted by the elaborate production and can imagine the way that they would want to produce it better. While others want to hear it pretty much the way it should sound and don't want to have to be too creative on their own.
If you don't sing or don't play guitar or piano, then you are going to have a tough time making even simple demos of your songs without spending money. If that's your situation, then I would highly recommend you team up with a great guitar player who can also sing, and you become a songwriting team and record simple demos of your songs together. Work together until you have enough really good songs to justify a day in a small recording studio. Then go in and record as many simple song demos as you can with the money you can afford for the studio, and use that demo to try to get some big publishing companies interested in signing you. If you get really lucky and they really like your songs a lot, they may give you an advance or possibly even pay to have some fully produced demos of the best songs recorded. Either way, at some point after you have a LOT of songs recorded in the simple demo form, you should pick out at least a few of the VERY BEST songs and find a way to get a professionally produced and recorded demo done with those songs... that way, you've got some really great sounding recordings of several songs to play for interested people, and then if they want to hear more of your songs, you can always fall back to the simple demo recordings you did of other songs.
Also, make sure you get LOTs of feedback on your songs through people that don't know you and can be honest with you. Find some free songwriting focus groups in your area (or start one if there aren't any) where songwriters get together on a regular basis and listen to and critique each other's songs. Don't be afraid to rewrite songs, and write as often and as much as you can... the more you work at it, the better you should get. Analyze the big hit songs and try to figure out what makes them special... really pay attention and listen to how everything comes together.
And, on top of all of this, get a real job!! You'll need to be able to make a living somehow while you are waiting for something to happen with your songs. There's a good chance that nothing will ever happen and none of your songs will every make any money for you. That's something you need to be prepared for. It doesn't matter how good you are because there are plenty of great songwriters out there with great songs... a lot of it is having the right people working for you to get your songs into the hands of people that matter, and having the right song at the right time (a lot of luck involved). All it takes is one hit song recorded by one very popular artist to generate a substantial income of royalties for you and really get your career moving.... but, be aware that the chances of that happening are about the same as your chances of winning the lottery.
On top of all of this, remember that I'm just one person and this is just my personal opinion on the matter. There are no right or wrong ways to do anything in this business, and sometimes it's all about making your own path to find your way into the business.