if you don't want subbass (80Hz downwards), you presumably want some midbass (80~200Hz). many factors contribute towards quality bass, including: headunit, amplification, speaker quality, installation quality.
the sony should be fine for the job, though the internal amplifier ICs are nothing flash in any headunit (sony or other). a proper amp would do a much better job. factory speakers are of course 'ordinary'. but if you improve the installation, it would help. though unsure if you'd be prepared to do a serious job: sound deaden the doors with self adhesive sound deadener (eg: Dynamat Xtreme), and cover over the open service holes in the doors. creating a 'sealed enclosure' for the midwoofers will significatly improve their performance, and the midbass.
so it's more than one factor - it's multifactorial.
lastly, i presume the sony has some form of equalisation, whether it be a 'bass' setting, and/or loudness setting (or similar). whilst you can dial in more bass, it will only push the speakers closer towards their mechanical limits (cone excursion), which can manifest as distortion at relatively low volume.