Officially, you should not be allow board your flight to the US without either your original I-551 (Green Card), or without an approved form I-131A which will require a trip to a US Consulate.
In "normal times" it may be possible to travel without the physical documentation if your airline already has the card on file, as you state occurred to your previously. This would generally only be possible if you were doing on-line check-in, and if the airline had sighted/scanned your card within a certain time period (every airline is different, but on at least some it's within the past 6 months).
However, times are indeed different at the moment. Before boarding, the airline will want to confirm that you are allowed fly to the US (most non-US citizens are barred from travelling there, with permanent residents being one of the exceptions that are allowed), and that you have a negative COVID-19 test result. This means that they are going to want to physically interact with each passenger, and check their relevant documents - and that's going to include your Green Card.
The odds of being allowed board a flight at the moment without being able to prove that you have the required documents to travel to the US (one of US passport, Green Card, visa or ESTA approval), AND without having the relevant documents that you are an exception under the COVID regulations (US Passport, Green Card, US citizen spouse, etc), is pretty much zero.
Attempting to have an unknown party transport your Green Card across country lines is likely asking for trouble. Almost certainly your best option is going to be to get it sent via courier (Fedex, UPS, DHL, USPS Global Express Guaranteed, etc).
