I don't have much of an issue with public domain works that are digitally hosted being recommended, but... I don't really see the point of it. If they're being hosted then odds are that we either already know they're good, or at least hold cultural value. Why bother recommending themwhen, most likely, they will not make onto to the list anyway?
There's already an issue with fics potentially getting updated after they are complete, potentially raising a derecommended fic to a better status or confusing the acceptance of listed and recommended fics by retroactively making them incomplete. AFAIK this hasn't happened yet, but a number of recommendations have already vanished entirely from their host sites, further muddying waters. It just creates more problems than it solves.
Regardless of your stance on webcomics being on the list,
Digger was initially paywalled, went free to read, was completed, got print published in six volumes, won a hugo award, got crowdfunded and republished, and is still available as free to read.
All of that happened because it is good. It got published because it was good. It's not good because it got published, which seems to be some of the thinking going on. That there is somehow an unfair advantage or something. There is so much complete shit out there that in theory went through multiple drafts and editors and was still published that, on consideration, I don't think there's much difference between fanfiction and original fiction.
Pre-internet stuff is contentious, and I think it should be. It wasn't made with format or accessibility in mind, and much of it is painfully dated. I think pieces that do make it onto the list would have to well and truly earn it, more so than most fanfiction already on there, by modern appeal and trend and evolution of writing having long since passed them by. Good on them if they make it.
That out of the way, I'd like to recommend two stories, starting with:
With All Awry, by Blair Bidmead. A Doctor Who(-ish) story, originally published in the charity anthology
Mythmakers Presents: Golden Years 1963-2013. This story is a sort of coda to Doctor Who's EDA novels, Faction Paradox, and
Scream of the Shalka, telling a transition from the slow death and autocannibalistic tendencies of the Wilderness years to the show's resurrection in nuWho. It is pure Doctor Who fanfiction,
and it is as canonical as Doctor Who/Faction Paradox can get, having had the same events directly portrayed/referenced from another POV during
A Romance in Twelve Parts. This is as much a piece of fan/fiction as it is a study of the Doctors trying to escape the narrative nonsense and implications of the cancellation/reboot. I suspect it is the kind of indulgence only very specific fans would appreciate.
And then we have
Canaries, by Dave Rudden. A Doctor Who story that loosely ties itself and
The Wintertime Paradox into
The Timelord Victorious mega event. It is a short story about Anke Von Grisel and her museum, The Verbier Museum of the Impossible, dedicated to things that shouldn't exist. It works extremely well as a stand alone piece, and I'd recommend it to anyone.