Can a third country national who is a legal resident of Portugal travel within the Schengen zone without a passport, just PT immigration card / ID?

Highly inadvisable, I would say.

Although there are no passport checks at the internal frontiers, as a third country national your passport is the guaranteed form of id and you should carry it together with your residence card for Schengen travel.

No doubt there will be people who say they cross the border into Spain regularly for shopping etc without any bother ever but even if I'm only intending a quick hop over, I carry both. If you want to stay in a hotel, it's always (as far as I know) passport that's requested. And if you're flying, airlines at airports outside of PT would almost certainly want to see passport for id at the boarding gate. The cartão de residência doesn't have the same international status within the EU as the bilhete de identidade.

Thanks; so what do you do when your passport is sent in for renewal?
That could take weeks, or months even.

In a word, no!

Fortunately I haven't had to deal with that particular scenario.

Despite what looked like posssibly daunting processing times, the last couple of renewals (of a British passport) were completed rather more quickly than I expected. In any case, I made sure to do the renewals at an appropriate time in the available window when I wasn't intending to travel. I can't remember for sure but I think there was a slight delay between submitting the application and being requested to send in the existing passport, after which the old (cancelled) and new passports came back fairly pronto under separate cover.

If I'd been in a hurry, I'd have investigated the possibility of an expedited service.

I don't know when I might need to travel. Sometimes something happens and I need to just go.

Also, I need the new passport back in time to apply for my next PT residence extension.

It probably all works just fine; but it seems nuts to have no passport for an unknown length of time.

Maybe you could book an appointment and travel to the issuing country to renew it in person within 1 day.

That's what the UK offers, for a fat fee.

In theory it should work fine; I've flown Schiphol-Alicante and returned Valencia-Schiphol using just my Belgian ID card ... though I did have my UK passport with me, but didn't need to produce it at any time.

Are you British?

I only ask because my very recent passport renewal (from Hungary) was done & dusted in 3 weeks.

No, I'm (gasp) American.
All passports are issued by a single specialized office of the state department, and sent by mail only.
Abroad, passports are sent to the consulate and they send them locally by post, although in person pickup is possible.

I have to send my current one in with the application.

If there is a US government shutdown, it could take a long time.

That's unfortunate!