In plain English
FAR 52.216-23, Execution and Commencement of Work, is a contract clause prescribed at FAR 16.603-4(b)(1), most recently dated Apr 1984. The complete official text is reproduced below, verbatim, from GSA's published FAR source files.
Its text contains no sentence requiring insertion into subcontracts; see the flowdown section below for what that does and does not mean.
Does it flow down to subcontracts?
No flowdown mandate found in the clause text
We scanned the full clause text and found no sentence directing the contractor to insert this clause into subcontracts. That is a statement about the text, not legal advice: a prime contractor may still flow terms down contractually, an agency supplement (DFARS, VAAR, …) may add requirements, and clauses listed below (if any) may order this clause into subcontracts from the outside.
Where it's prescribed
As prescribed in 16.603-4(b)(1), insert the following clause in solicitations and contracts when a letter contract is contemplated, except that it may be omitted from letter contracts awarded on SF 26 :
Prescribing reference: FAR 16.603-4(b)(1).
The official text, verbatim
As prescribed in 16.603-4(b)(1), insert the following clause in solicitations and contracts when a letter contract is contemplated, except that it may be omitted from letter contracts awarded on SF 26 :
Execution and Commencement of Work (Apr 1984)
The Contractor shall indicate acceptance of this letter contract by signing three copies of the contract and returning them to the Contracting Officer not later than _______________ [insert date]. Upon acceptance by both parties, the Contractor shall proceed with performance of the work, including purchase of necessary materials.
(End of clause)