The business model with the distribution centers works very well even though it might not make any sense to the rest of us. The logistics and practices enhanced/developed by FedEx in particular have changed the way business is done. Also, the logistics and centralized store houses combined with outside vendors is what Amazon/Jeff Bezos did so well and why that has changed how we buy/get things.
Bear, without getting overly political I can say that I started noticing changes in the post office this past June. I think you would be fine with a standard overnight mailer -- when you use things like the flat rate boxes, tracking is included on your receipt. Back in June although prices on flat right boxes didn't change, there was fine print on the receipt that amounted to "by the way, we are not honoring the guarantee of the extra money you are paying for 3 day shipping will actually happen and it might be 5-7 days". I have been shipping many flat rate express packages and they are mostly (but not all) arriving on time.
Short form (too late!): I think if you do your due diligence on your end and mail in time (so if you mail overnight with 3 days left to get to them), even if the post office is delayed, your check would have to be honored.
Many places now are doing things on-line/direct transfers etc. that never did before. Our Washington State Driver's License agency has a big notice on their page that they didn't want me coming in, even if my license was expired (which it was), even if I had gotten the extension last time (which I did - means I should have gotten vision check and new picture), just click this button here and send them money and they would send me a fresh license.