Hmm...my understanding was that the Ppt stayed on until a new one was chosen if there was no VP. Looking at our actual instances, Crawford held the position for the entirety of the gap between Clinton's death and Gerry's inauguration; Gaillard for the whole period from Gerry's death to Tompkin's inauguration; White after Calhoun resigned; Southard then Mangum after Harrison died; King, then Atchison after Taylor's death; a bunch of people, but always someone, after King died; Foster, then Wade after Lincoln's assassination. I don't see any real gaps, although I might be missing something.
The problem in 1881 looks to have been that there wasn't a sitting president pro temp when Garfield died, apparently because Arthur had been presiding personally throughout the special session. Congress didn't reconvene until a couple weeks later, so there were a few weeks without a Ppt. But that would've happened in the event that there'd been a provision for the Senate to choose a new VP, too, since the problem was that the Senate wasn't in session when the President died, not that the president pro tempore's position had lapsed due to the Senate no longer being in session.
Also - while the senate can of course elect and remove a president pro tempore at will, I'd assume that any president pro tempore who actually succeeded to the presidency would pull a Tyler and assert the full powers and his independence of any further Senate control. Interestingly, though, the 1792 Presidential Succession Act actually provided for a special election within the year if both the president and VP died. The Special Election was only removed when the 1886 act diverted succession to the Secretary of State.