I am not an expert of polymerizations, but actually free radicals are the ones that keep the reaction going, not the ones that make it 'end too quickly'.
The kinetic of chain polymerisation requires a slightly different mathematical treatment with respect to 'normal' reaction kinetics.
However, the process starts with the formation of radicals by homolytic cleavage of an initiator I (sorry for the poor formatting):
I --> A. + B.
This occurs by supplying energy (heat, radiation...), but I is designed to undergo this reaction easily, so it's not a problem.
The monomer R-H reacts with the initiator radical:
R-H + A. --> R. + AH
The monomer radical can then add to other monomers to generate a longer chain (which is the aim of the process):
R. + R-H --> R2.
R2. + R-H --> R3.
etc.
Or it can abstract a hydrogen atom from another molecule (chain transfer):
Rm. + Rn-H --> Rm-H + Rn.
As you see, in all these cases a radical is present on either side of the reaction arrow, which means the reaction can proceed (as long as there's monomer available).
The cases when the radicals are quenched are:
- the addition of two growing chains:
Rn. + Rm. --> R(n+m)
- the reaction with labile substances that yield very stable, unreactive radicals (X.):
Rn. + X-H --> Rn-H + X.
I seem to remember that as there are initiators there are also other substances one can add to moderate the reaction; please don't ask me what they are (phenols? thiols?).
But normally the result of your polymerization (mean molecular weight, distribution of molecular weights...) is determined by the initial amount of initiator and by the specific kinetic constants of the system.