Shouldn’t a double sided S1-1/School Area sign be manufactured to be mirrored (each opposing side reflecting the mirror-opposite of the other as viewed by oncoming traffic)?

No, the School sign (S1-1) does not need to be mirrored in a School Crossing Assembly. The arrow plaque (W16-7P) needs to point to show the location of the crossing, whether it is to the right  or to the left.

Section 7B.12 of the MUTCD (part):

Section 7B.12 School Crossing Assembly

Standard:

01 If used, the School Crossing assembly (see Figure 7B-1) shall be installed at the school crossing

(see Figures 7B-4 and 7B-5), or as close to it as possible, and shall consist of a School (S1-1) sign supplemented with a diagonal downward pointing arrow (W16-7P) plaque to show the location of the crossing.

02 The School Crossing assembly shall not be used at crossings other than those adjacent to schools and those on established school pedestrian routes.

03 The School Crossing assembly shall not be installed on approaches controlled by a STOP or YIELD sign.

Section 7B.12 of the NYS Supplement to the MUTCD (in part):

Section 7B.12 School Crossing Assembly

Guidance:

03A The School Crossing assembly should be installed at marked crosswalk(s) used by students going to and from school (see Figure 7B-4), as determined by an engineering study. The School Crossing assembly should not be installed at signalized intersections.

Here is a figure from the National MUTCD that shows this.

https://mutcd.fhwa.dot.gov/htm/2009r1r2/part7/fig7b_06_longdesc.htm

Resources:

MUTCD:
https://mutcd.fhwa.dot.gov/pdfs/2009r1r2/pdf_index.htm

NYS Supplement to the MUTCD:
https://www.dot.ny.gov/divisions/operating/oom/transportation-systems/repository/B-2011Supplement-adopted.pdf