If the new thermostat does not require a neutral then no, you don't need to connect a neutral to it. However you do need to make that conductor safe rather than just sticking tape over it!
You can't just do away with the earth as the regulations require that it be present at every pount of termination, the obvious issue here is that cable doesn't have an earth in it, so you need to assess whether a class 2 thermostat will maintain the required level of safety or whether something else needs to be done.
It's used to hone the switching temperature hysteresis, so it's not needed but better being there. Although if that bottom pic is the new on, there is no connection for either an earth (CPC) or neutral.
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