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
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.
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.
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.
Well, the non-metals and Manganese are way more available than iron anyway (probably molybdenum too). But it will be really difficult to create high-quality steel.