Do banks make profits by selling bonds to the central bank?
No, banks don't directly profit from selling bonds to the central bank. These bonds are sold at face value, there is no discount or premium, this is not part of the system.

My best guess of what the author means is that by selling bonds to the central bank, commercial banks get cash, which they loan out and make a profit on the loans. So commercial banks profit from selling bonds to the central bank in an indirect way. Attributing banks' profits from lending to the central bank is a bit of stretch though and not a very appropriate statement in my view, so I fear I might have misunderstood the author.

What the article seems to promote is already a reality in communist China, where state-owned banks lend at favourable rates. There can be significant problems with this approach of cutting out private banks, as the Chinese bad loan problem shows (although private banks can also accumulate bad loans, but maybe less so due to competition), but this is a different discussion.