Ellis Island has a Samuel Roscoe, aged 4, (actually 3 yrs 6 months), coming through Sep 20, 1902. His mother, Anne, 48, is a housewife, they are US citizens, last residence was Chorley, England, and they are headed to Joseph Roscoe, Anne's husband, in Pittsburgh, PA.
There are lines through this entry, so they didn't actually take this ship in the end. They must have caught another one which came via Boston instead.
Possibly the reason Samuel David Roscoe can't be found on the freeBMD indexes is that he was born in the US! Perhaps his mother took him back home when he was quite young to meet the family, and he only remembered the trip back, and that's how the family story got started.
A Joseph Roscoe m. Dec 1888 quarter possibly an Anne Baldwin in Chorley. There is also a Jos. Roscoe with wife Ann in Chorley in 1881, she is 27 which would make her 48 in 1902; this could be them.
This family emigrated in 1882; Ann Roscoe, aged listed as '59' (must be wrongly transcribed, presumably '29'), is on the Arizona, arrived 18 Sep 1882 into Castle Garden with children Elizabeth E, 4, Peter, 3, Mary, 2, and Ann, 'infant'.