What are the advantages of applying for Medicaid at the DFS office over applying through the nursing home?

The nursing home is a business, looking out for its bottom line.

Medicaid is less apt to tell you something like "oh, it'll be fine, just sign here as being financially responsible".

An eldercare lawyer is looking out for YOUR interests.

Applying thru the NH is essentially adding a middleman to the process. I applied directly at the Medicaid office, was assigned a caseworker, was put in Medicaid pending status immediately. The caseworker was the only person I had to deal with, she and I spoke frequently, and I faxed her documents directly. It was a very easy process, only hampered by my inability to find Moms marriage license and divorce decree and the status/whereabouts of her exhusband/my father. The caseworker finally came up with a workaround that would pass muster with her bosses. This could have taken even longer if I had the NH as middleman in the process.
One thing I do caution you about however you decide to apply, is to NOT make the NH the payee to any of Moms SS or pension checks. I have all moms income direct deposited into her checking account, get a monthly bill from the NH for her share of cost, and write a check. That way I know exactly everything is working correctly and it only take me a minute. My GF had her Moms income go directly to the NH, and was in ignorant bliss, until one day the NH told her she owed $60,000. Evidently her pension administration thought she had passed away and stopped paying the NH for months. It took a Long Time and tons of work by GF to get this corrected. If you’re in charge of the money, you can avoid surprises.

I agree with rocketjcat. Go to Medicaid directly.

I had no problem in making the NH payee of Moms SS and pension. Don't understand how the pension screwed up, I had to send a death certificate to prove Moms passing. I liked that I didn't have to worry about that. And in the end, they had to deal with SS if SS withdrew money too soon. It was the financial offices fault for not contacting the POA if the pension stopped. Medicaid would have adjusted their amount if for some reason the pension stopped legitimately. Someone didn't do their job and auditors should have picked it up.