First to the Ring Gear.
Most people do not realize that a proper Ring Gear appears totally damaged the day it leaves the factory. If the teeth were cut square and flat like a normal gear, the starter pinion would often jam against the face of a Ring Gear tooth and not fully engage between the Ring Gear teeth. Manufacturing requires that each tooth of the Ring Gear be cut and angled so that the starter pinion easily inserts left or right and fully engages the teeth. If you think yours is damaged, please post a pic and let others opine about the condition. I have seen many misinterpret Ring Gear condition.
Second for the starter.
You have to imagine in slow motion. The solenoid, when activated, has two somewhat separate functions. The first is a magnetic pull to move the internal plunger. Through leverage, this motion thrusts the starter pinion shaft forward to impact the Ring Gear and fully engage the pinion teeth with the Ring Gear teeth. It is important that the starter should NOT be spinning during this insertion action. If the pinion were already spinning before injection it would surely tear up the Ring Gear teeth or its own pinion teeth. Once the pinion gear is fully inserted into the Ring Gear then the outboard end of the solenoid creates a heavy electrical bridge to feed power into the starter itself and begin the spinning action. This sequence of timing the insertion and the spinning is critical. The timing is created or maintained by gaskets and/or spacers where the solenoid rests into the starter motor body.
You have a 'click' but no spin. Can you determine if the click is coming from the starter relay? Or is it coming from the body of the solenoid?
A weak battery can create these symptoms. How old is your battery? Have you tested its load voltage? What class of battery technology is it? AGM class is preferred as they tend to have a slightly higher standing voltage and that makes all electrical things work better and faster.
Ultimately an old solenoid can have this heavy electrical bridge corroded or pitted. The solenoid can be disassembled to resurface and recondition the bridge. Did you include a new solenoid with the new Valeo starter? Or did you utilize your old, original solenoid installed into the new Valeo starter?
Further reports or pics would be helpful.
Patrick Hayes
Fremont CA