No. There is no good reason to field cure specimens as there is no correlation between the in-place conditions and the small specimens taken for compressive strength. While field curing is often done, it gives one a false sense of either acceptance or rejection criteria.
Here are two examples:
Suppose the ambient site temperature is 60F. You cast 4 cylinders in the field for field curing, to represent two interior columns and two interior beams, each having 24"x24" cross sections. The concrete is delivered to the site at 70F.
The field cured specimens will reach equilibrium at 60 degrees in a few hours. The larger mass of the structural members will not reach equilibrium for several days because of the exothermic hydration reaction in the concrete.
The field cured specimens are tested and show lower than expected strength because of the lower curing temperature. Form release is delayed, causing extra cost.
Now the other example:
Ambient site temperature is 90F. Concrete delivery temperature is 75F. Field cured specimens reach equilibrium at somewhere between 75 and 90 degrees in a few hours. Structural members reach equilibrium temperature in a few days, but it is less than field cured specimen's temperature. FC specimens are tested and result shows higher than needed strength. Forms are released. Concrete in structure not as strong as FC specimens indicated....problems.
The specimens are to check the mix design, not the in-place concrete. While one might think it is a stretch to base the entire design on the mix design properties, it is more valid than trying to account for the numerous variables encountered for site conditions.
The design is done. The basis of the design is f'c. The mix design is to represent f'c and the specimens are to represent the mix design.
If you want to know the in-place variation in the concrete, then use a variety of methods...all of which must be correlated, with the exception of cores. Common methods are ultrasonic pulse-velocity, Lok-Test (pullout) testing, or cores. Methods such as the Swiss Hammer and Windsor probe have some minor validity in evaluating concrete, but in my opinion, are never valid for the acceptance or rejection of concrete based on their indications of compressive strength.