The problem is, when would we get an answer? Even in the best case, that would be 240 years. But we don't have the capability of building probes which approach

The problem is, when would we get an answer? Even in the best case, that would be 240 years. But we don't have the capability of building probes which approach the speed of light, so with current technology, it is probably hundreds of thousands of years. So the short term solution (in the order of decades) is improve the observation capabilities. More observations with the JWST should give a better idea what the next iteration of a space telescope for this job should be and give some motivation of accelerating building such a telescope. JWST itself took decades to build and deploy. It would be remarkable, if an improved telescope could be deployed in a 10 year time frame. Of course, an operational Starship would help a lot with that, as a lot of effort for JWST was the complex deployment mechanism.