You oversimplify what really happened to the standards of rigor. Rigorous proofs
were invented by ancient Greeks. (According to their own tradition, by Thales, but his writings do not survive). At the mature stage of Greek mathematics (Euclid, Archimedes, Apollonius) a very high standard of rigor was established.
It changed little since then to the modern time, if one speaks of the mainstream
mathematics (excluding the "new" areas of mathematical logic, set theory and foundations).
With the invention of calculus, this standard declined, because of the lack of foundations of calculus. Still at the time of Newton, best mathematicians perfectly understood this ancient standard; they just could not expose calculus on the same high standard as the Greek geometry.
And I suppose this is the main reason why Newton preferred to use
classical geometric methods (rather than calculus) in his Principia.
In 18th century people were more concerned with discovery of new things and in applications, then in the standards of rigor. So the standards declined. But in 19th century, by a long and difficult gradual process, this standard was finally achieved again. Rigorous foundation of calculus emerged.
In the end of 19th century, the attention to foundations increased. This followed by a crisis in foundations in the beginning of 20th century, until some satisfactory situation was finally established. Again, this crisis in foundation
did not really affect much the work of most mathematicians.
Those mathematicians who are not interested in foundations write on approximately the same standard of rigor which was established by the Greeks, leaving the care about foundations to the specialists. (This is somewhat oversimplified picture: there is some important interaction between foundations
and the rest of mathematics, but I cannot go into detail here).
In all epochs, mathematicians were sometimes more concerned in discovery of new things, then in rigorous justifications. Archimedes himself wrote a famous book
with heuristic discoveries. He perfectly understood that the rigorous proofs were lacking. In the modern times, there is also plenty of such kind of research, I mean of the boundary of mathematics and physics. Where people use
"semi-mathematical" reasoning to discover new things, and don't know how to
prove these things rigorously.