1. Your link says these are elements commonly found in steel, not that they are all required. In fact it says of phosphorus and sulphur that they are generally undesirable.
2. We don't need to make a steel sword, an iron sword could do.

Either way you would definitely need carbon, but as you say that's pretty easy. I don't think any of the other elements are absolutely required.

Steel requires only iron and up to about 2% carbon

Rest are minor alloying elements used mainly in modern steel alloys to improve the steel beyond what just carbon steel could do like for example stainless steels

Those are all of them, but that's for a lot of different types of steels. You don't have to have all of those metals to make steel. You really just need iron and a tiny bit of carbon. A few of your ingredients help with purity, and the rest are additives for different steel properties you may want. Like a touch of nickel for stainless steel.

You only need iron and carbon the rest is already alloyed steel. You can definitely make a good blade out of only iron and carbon, it won't be stainless, it might be difficult to harden just right, but it will be flexible and hold a keen edge if forged right. The smiths of ole dealt with nastier steels containing all kinds of things making it worse, not better (such as excessive amounts of sulphur and phosphorus) so I'd say they'd manage.