A list of these long period (>200 year orbit but <1000 year) comets is here.
https://en.wikipedia.org/wiki/List_of_long-period_comets
One can define "come close to the inner solar system" as less than the distance of the asteroid belt (~3 AU). I count ~100 of these comets with a perihelion of <3 AU out of the list of 116. 36 of the comets on this list pass less than 1 AU from the Sun.

Thus, I do not think your statement is correct.

The time frame of a mere 200 years seems infinitesimally short in the long time of the solar system (5 billion years), so they should have passed millions of times through the inner solar system over that period.

Also look at this list.
https://en.wikipedia.org/wiki/List_of_near-parabolic_comets
One of those has an orbit period estimated to be 130,000,000 years! That one would have passes as close as 4.2 AU ~38 times over the solar system age.