I only skimmed the article, but I don't see where it gives reasons why solvency matters, although it hints it might. I was under the impression it doesn't really matter if the fed is solvent, since I can't think of any concrete consequences.
Having inflation above the interest rate helps decreasing the debt/gdp ratio.
I mean, that might not be how things are set up right now, and it would make their position even more negative since newly-created cash is a liability on their balance sheet and they didn't get an asset for it. So maybe at the moment it's on the treasury to make up the difference. But I don't see a fundamental reason why that needs to be the case, it seems incidental to the current setup.
And yeah, even under the current system the treasury could sell bonds to fund the shortfall, and then the Fed could just buy those bonds (via the market) and keep them on their balance sheet, rolling over forever and so effectively loaning the money to the treasury at zero interest forever. So that's just the same thing but with extra steps.