They’re not really the same thing.

ETTR is a capture-stage exposure technique: you expose as brightly as possible without clipping important highlights so the sensor records the strongest possible signal, especially useful when shooting RAW. Its benefit is better use of the sensor’s available data and potentially cleaner shadow recovery.

A LUT is not a capture method. It’s a mapping applied to already-recorded image data, usually for preview, display, or grading. Likewise, log/HDR-style encoding affects how recorded tonal data is stored or displayed, but it does not create detail that the sensor never captured.

So:

• ETTR advantage: can improve captured signal quality if exposure is increased without clipping.
• ETTR downside: if pushed too far, highlights are lost.
• LUT/log advantage: gives a more flexible or convenient tonal rendering for viewing/post.
• LUT/log downside: cannot add dynamic range or recover clipped data after capture.

They can be used together, but for different reasons: ETTR helps optimize what the sensor captures; LUTs/log help interpret or grade that capture afterward. Combining them is not “double HDR” or extra dynamic range by itself—it only helps if ETTR improves the original capture and the later mapping makes better use of that recorded data.