Contactor control with solid state relay

Hello! I do not think that the type of relay will change the life of the contactor coil. However, this will eliminate the influence of the solid state relay. The only question remains, what are you protecting the windings from? What voltage are you feeding the coil? What is the mains voltage? In the normal case, when the coil is energized with rated voltage, it should not burn.
There are two options left:
1. The problem is related to the quality of the coils.
2. Problem associated with difficult movement of the starter armature - check the movement of the magnetic system.

By the way, the BroadLink SC1 WiFi relay can withstand a current of up to 10 Amperes, which allows you to power any of the electromagnetic starters or contactors I know of. Try your system with a different starter. And by the way, the output of such wi-fi relays, as far as I know, is relay, which is ideal for your tasks.
P.S.
From the characteristics of the starter - the rated voltage of the control circuit Us, AC 50Hz 184... 253 V, which suggests that the coil should not burn out even at increased voltage.