See, for instance, A. A. Michelson's claim in the second lecture of his 1899 (pub. 1903) lectures Light Waves and Their Uses:
Before entering into these details, however, it may be well
to reply to the very natural question: What would be the
use of such extreme refinement in the science of measurement? Very briefly and in general terms the answer would
be that in this direction the greater part of all future discovery must lie. The more important fundamental laws and
facts of physical science have all been discovered, and these
are now so firmly established that the possibility of their ever
being supplanted in consequence of new discoveries is exceedingly remote. Nevertheless, it has been found that there
are apparent exceptions to most of these laws, and this is
particularly true when the observations are pushed to a limit,
i. e., whenever the circumstances of experiment are such that
extreme cases can be examined. Such examination almost
surely leads, not to the overthrow of the law, but to the discovery of other facts and laws whose action produces the
It is unclear to me whether this was merely an unpopular, optimistic bluster, or whether this quote represents an entire school of thought at the time. Especially troublesome to me is Michelson's involvement in the famous 1880s experiments on light and ether... surely the conflicting results gave him some pause?
On the other hand, note his last two sentences above - Michelson is clearly not positing that all of physics is "done" per se. Instead, his view seems to be that "the more important fundamental laws" are correct, modulo certain small improvements that can still be made.
Perhaps it is this bolded statement that represented the general attitude of the time, and not the statement that physics was "finished." Of course, while this quote does not conclusively answer your first question, it does provide another data point towards the second.