Why is it runway "1" instead of "01" in America?
Indeed USA deviates from ICAO:

5.2.2.4 Zeros are not used to precede single-digit runway markings. An optional configuration of the numeral 1 is available to designate a runway 1 and to prevent confusion with the runway centerline.

Source: USA AIP - GEN 1.7 Differences From ICAO SARPs
5.2.2.4 in ICAO SARPs Annex 14 (Aerodromes) is as follows (emphasis mine):

5.2.2.4 A runway designation marking shall consist of a two-digit number and on parallel runways shall be supplemented with a letter. On a single runway, dual parallel runways and triple parallel runways the two-digit number shall be the whole number nearest the one-tenth of the magnetic North when viewed from the direction of approach. On four or more parallel runways, one set of adjacent runways shall be numbered to the nearest one-tenth magnetic azimuth and the other set of adjacent runways numbered to the next nearest one-tenth of the magnetic azimuth. When the above rule would give a single digit number, it shall be preceded by a zero.

Shall in ICAO legalese denotes a standard.
The aforementioned optional configuration for &quot;1&quot; covers your concern regarding the centerline:


The numeral &quot;1&quot;, when used alone, contains a horizontal stroke, as shown, to differentiate it from the runway centerline marking.

Source: FAA AC 150/5340-1
The rationale is not in the AIP; it's not always given. Deviations from SARPs are also very common, that's why there's an AIP section dedicated to them.