Looks like the reason this isn't converting to line feature directly is because it's not a KML LineString (how standard line features are usually encoded in KML). If you unzip the KMZ and look at the doc.kml file inside, you'll see that it contains a Folder with 28 PhotoOverlay features, one Point Placemark for the start of the trip, one Placemark containing the Track (animated line), and one Point Placemark for the end of the trip. So when you convert using GDAL, the only normal geometries it's finding are the two points.
Sounds like you found a via geojson and QGIS to convert the track to a Line, but you say that it lost the attributes. What attributes are you looking to retain? The Track has time-stamps for each vertex, as well as attached speed data at each vertex. But when converting it to a Line there's no way to apply those times or speeds as attributes to each vertex. If you wanted to save those you would need to convert the track to a series of points instead of a line.
The only other "attributes" I see are all in a single HTML blob in the description tag for the Point at the end of the trip. You could manually parse those out into attributes if you wanted.
So, using some find/replace, regex and a bit of manual cleanup, below is a LineString version of your KML Track, with the Description contents broken out into ExtendedData name/value pairs. That should convert to a Line with associated attribute table. You can copy/paste it into Earth to see it, or save as KML to import/convert it.
If you have lots of these to do, you might want to look at GPSBabel, which is a nice tool with both a UI and Commandline interfaces for all sorts of format conversions, and looks like it's able to convert KML Tracks into LineStrings.
<Placemark>
<name>
<![CDATA[Tuliza Trip 1]]>
</name>
<ExtendedData>
<Data name="Name">
<value><![CDATA[Tuliza Trip 1]]></value>
</Data>
<Data name="Activity_type">
<value><![CDATA[driving]]></value>
</Data>
<Data name="Description">
<value><![CDATA[-]]></value>
</Data>
<Data name="Total_distance">
<value><![CDATA[4.45 km (2.8 mi)]]></value>
</Data>
<Data name="Total_time">
<value><![CDATA[46:00]]></value>
</Data>
<Data name="Moving_time">
<value><![CDATA[46:00]]></value>
</Data>
<Data name="Average_speed">
<value><![CDATA[5.80 km/h (3.6 mi/h)]]></value>
</Data>
<Data name="Average_moving_speed">
<value><![CDATA[5.80 km/h (3.6 mi/h)]]></value>
</Data>
<Data name="Max_speed">
<value><![CDATA[25.26 km/h (15.7 mi/h)]]></value>
</Data>
<Data name="Average_pace">
<value><![CDATA[10:20 min/km (16:39 min/mi)]]></value>
</Data>
<Data name="Average_moving_pace">
<value><![CDATA[10:20 min/km (16:39 min/mi)]]></value>
</Data>
<Data name="Fastest_pace">
<value><![CDATA[2:22 min/km (3:49 min/mi)]]></value>
</Data>
<Data name="Max_elevation">
<value><![CDATA[22 m (72 ft)]]></value>
</Data>
<Data name="Min_elevation">
<value><![CDATA[1 m (4 ft)]]></value>
</Data>
<Data name="Elevation_gain">
<value><![CDATA[37 m (123 ft)]]></value>
</Data>
<Data name="Max_grade">
<value><![CDATA[5 %]]></value>
</Data>
<Data name="Min_grade">
<value><![CDATA[-12 %]]></value>
</Data>
<Data name="Recorded">
<value><![CDATA[10/18/2014 07:25]]></value>
</Data>
</ExtendedData>
<LineString>
<coordinates>
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