I have an amusing anecdote:

I called my local SSA office the other day to ask on procedure for replacing my "Valid Only with DHS Authorization" annotated Social Security Card with one that does not carry the caveat, now that I have my Green Card (side note: had for over 6 months, and still look at it every day...). After a little to-and-fro with the rather rude gentleman on the other end of the phone, who insisted that an appointment for such a request was not a valid reason for coming to the office (despite their website saying it was!), he then instructed me to mail both my original passport and original Green Card to the office instead with my existing SSN card, as even notarized copies are not accepted.

OK, so yeah, you're asking me to entrust my much prized identification documents, to a service which has proven to have (in our neighborhood anyway) tremendous difficultly delivering mail to the correct addresses written on the envelope. I'm sorry that'll be a "no". He even stated that, if I decided to send it with a more trackable mail service, the same care and compliment would not be returned when they mailed the documents back and if they get lost, then "that's not going to be the SSA's problem".

Good job I don't actually need the SSN card if/when I need to fill out a new I-9.

It is not a requirement to present the actual SSN card and so why do you need this now? You can always claim that you lost it and get a new one once the pandemic is over.

Oh, just vanity. Nothing more!

Claiming you lost it would likely result in being issued the same one with the restriction on it, what the OP is talking about is getting an unrestricted one once you have a green card.I have always asked for it when adding a new employee to payroll and in practice most have presented it anyway when doing the I-9 because you mostly do it all together anyway, but you are right, it can also be verified online if not available for presentation:
https://www.ssa.gov/employer/SSNcard.htm

I had a very similar conversation with someone from the national SSA customer service center. The SSA claims that their offices are shut due to COVID-19, but somehow DMV offices, USCIS offices, local county offices, and many other government offices have managed to stay open during this pandemic. What's so special about the SSA?

I had a similar situation when applying for Medicare. I sent my documents by registered mail and was actually assured by the person I spoke to that the documents would be returned by registered mail. They sent them back by regular mail...