Is there any realistic situation where a court decision might hinge on
producing the original physical paper certificate?
Yes. But there are alternative options to prove the same thing, even then.
A copyright registration certificate is not like a dollar bill or an original promissory note, where a copy isn't as good as the original, even though it can be slightly convenient to have the original in hand.
Or are they just souvenirs that I can safely shred?
If you scan them and then shred them, nothing horrible will happen. In the rare worst case scenario where the fact of your registration is disputed by a stubborn defendant in a copyright infringement lawsuit all of the way up to trial, you could get a replacement certified copy from the Copyright Registrar, or you could prove your registration with other evidence.
Registration is not required for a copyright to come into being in U.S. law. 17 U.S.C. §§ 102(a) and 408(a). But it is a condition precedent (i.e. a requirement that must be met prior to taking action) to certain legal remedies. 17 U.S.C. § 412. So, proving your registration is usually one element of your infringement claim in a typical copyright infringement lawsuit.
In that lawsuit, if you actually go to trial, you need to prove that the copyright was registered in a timely fashion, if the other side won't stipulate to that fact.
But it would be rare for a defendant who had been presented with a copy to insist that the fact of registration be proven at trial because being that stubborn is likely to piss off the judge unless they can successfully prove that you have faked your claim to have a registered copyright, which is something that almost no intelligent lawyer in a copyright infringement lawsuit wants to do, since judges have broad discretion on so many issues, even in a jury trial.
The easiest way to prove the fact that your copyright was registered on a particular date is to offer up a certified government document like the ones that you have in your office into evidence as an exhibit (typically, one of your very first exhibits in your infringement case). This is because an official government certificate is exempt from the hearsay rule, is "self-authenticating" ,and its contents are admissible if not barred by another rule of evidence (which it almost never would be). The legal reason for keeping these paper documents is to get that evidentiary benefit at trial. See Federal Rules of Evidence 803(8) (exceptions to the hearsay rule), 902 (self-authentication) and 1005 (admissibility of public records).
Of course, mostly, these fancy certificates are issued as a matter of pride and ego and customer satisfaction. Prior to an actual trial or evidentiary hearing in a copyright infringement lawsuit, a copy, supported by your declaration under penalty of perjury that the copy isn't a fake, would be the usual way to establish your registration in documents that are e-filed through the federal electronic e-filing service called CM/ECF rather than presented in person on paper anyway.
Typically, pretrial document disclosure is done via an electronic transmission of scanned copies as well.
In practice, it would be quite rare for a defendant in a copyright infringement lawsuit to insist that you prove up your registration at trial, rather than just stipulating to it. Indeed, it is you who might not want to stipulate in a jury trial, just so you can flash an important looking document in front of the jurors right at the start of the trial for dramatic effect. But you don't have to do that.
Ultimately, if the defendant really did insist on proof of your registration, you could get a new certified copy from the Copyright Registrar (which wouldn't be quite as pretty, but would have the same legal effect) for a small fee without too much advanced notice, or you could introduce your scanned copy into evidence and authenticate it through, for example, your testimony on the stand under oath that you scanned the original and that it is a true and correct copy of the original.
You also might want to keep a paper copy because it will still be around if all of the relevant electronic records are destroyed or inaccessible for some reason. But if that happens, copyright infringement may be the least of your worries.