Replacing a broken control with a new one does not determine how a product is affected by the Ecodesign regulation. If the local space heater (separate control+ heater) was placed on the market before the date of application of Regulation (EU) 2024/1103, the Ecodesign requirements applicable to that product at the moment of its placing on the market will still apply to that product after the replacement of the control.
Moreover, replacing a broken control is a pure maintenance operation. Nothing prevents the user to replace the faulty control with the same type of device, not existing any obligation to fit a better performing equipment in order to fulfil the legislation on Ecodesign. Where this be the case, it should be explicitly set out in the regulation, which is not the case.
Disclaimer: Please note that the European Commission cannot provide a legally binding interpretation of the EU legislation, as this is the sole competence of the European Court of Justice. Any remarks from the European Commission services are without prejudice to the position the Commission might take should related cases arise in a procedure before the Court of Justice.