The 20th Century Limited was discontinued before Amtrak but there was other NYC/PC service on the route until Amtrak Day. The Lake Shore Limited route was not included in the initial basic Amtrak system and there were a few days at the start of Amtrak with no service. Service was returned (as the "Lake Shore") between Chicago and New York in 1971-1972 with some state operating support. Then was restored in 1975 (as the "Lake Shore Limited") between Chicago and New York/Boston.
EDIT: So, to directly and more clearly answer the question, there was a short gap in service on the route in 1971 and then again from 1972 to 1975.
Did Amtrak run the Empire Service at the time? Were the Empire Service levels similar to today's?
I (possibly incorrectly) remember that the commuter service on the Hudson Line prior to the formation of Metro-North was simply horrible: bad timekeeping, unwashed cars, no through trains past Croton-Harmon.
Amtrak ran service to Buffalo/Niagara Falls from day one.
Indeed PC Commuter Service was never better than horrible in most cases. There were minor exceptions I am told.
Amtrak ran as far as Buffalo (the old Central Terminal) from the beginning, but as was mentioned in Eric S's post above, there was that "gap period", when that was as far as Amtrak ran.
Amtrak did have a cross-platform connection with a train, usually Rail Diesel Cars, operated from Buffalo to Toronto by Penn Central (later Conrail); Toronto, Hamilton, & Buffalo; and Canadian Pacific, via Fort Erie, Welland and Hamilton.
Later, besides the 'Lake Shore' train, Amtrak extended the Empire State Express from Buffalo to Detroit. Later, they renamed it the Niagara Rainbow, but it did not actually go thru Niagara Falls until October of 1978, and then it ended service altogether in 1979. Amtrak used the name in the '90's for a briefly operated once a week overnite train to Toronto.
In 1981, Amtrak started thru service together with VIA Rail from New York to Toronto, via Buffalo Exchange Street, and Niagara Fall, NY and Ont, then the CN route to Toronto.
It is interesting to note, that when the Lake Shore was extended by funding from New York and Ohio, Pennsylvania declined to contribute, and as a search of old TT's show, the train did not stop in Erie...