Cleaning up layer by deleting features with only one overlapping vertex in QGIS

You can achieve the result that you are striving for with GRASS toolset v.clean for cleaning topology of a vector element. Probably, you will need to deploy rmdangle option.

rmdangle: remove dangles, threshold ignored if < 0

Important to set up Threshold and snap tolerance parameters.

Check the following references for more details:

• Removing 'tails' from line features?
• Deleting lines in QGIS
• Automated removal of dangles / loose ends in line with QGIS?