I would check that the voltage regulator is grounded to the engine. According this engine uses the tri-circuit stator with a two wire voltage regulator [one in and one out] or the dual circuit stator.. Many time Briggs will mount them on the plastic parts and run a ground wire from the mounting screw to the engine. Also it output voltage with engine at 3600 rpm it should be above the standing battery voltage [voltage measured with the engine not running] but less than 16vdc. If the standing battery voltage and regulator output voltage are the same then the regulator mostly bad; although, a stator is possible [just not as likely]. To check this check the input lead of the voltage regulator [stator output lead] and it should be near 40 vac.
The dual circuit stator can be used on this engine and it does not use a regulator but instead as diode in one of the leads. These tend to burn out [opens up] and when they there will be no output voltage other than the standing battery. When working the output can be from slightly above the battery standing voltage to 16 vdc but around 13-14 vdc. On this stator Briggs does give any measurement other an amp of up to 3 amps on the DC lead for the battery and Ac voltage of 14 vac on the lead for the lights.
Wish be exact on which stator setup Briggs but they generalize the IPL.
Just note the Ariens wiring diagram would suggest the tri-circuit was used originally.
Here is the wiring diagram for the Ariens.