Actually no, a headless design is inherently more pitch-stable:
Far less string length outside the vibrating length, so less strechy, and less (non-frictionless) slipping back and forth over the nut.
String is completely straight past the nut. Winding a string around a tuner post creates a string with more 'give' and 'springiness'.
The effective 'ratio' of the tuner (assuming standard headless linear-pull type) is higher, so tuning is more precise.
Not being able to hang it is not an issue, hanging a bass from its head is not such a good idea anyway (for obvious reasons: unnecessary stress on neck and neck joint, and neck is not free to settle into its natural shape). That is just something that happened due to headstocks existing.
Clip-on tuners are inherently inferior, you are better off without them (note: you cannot use them for intonation).
'Dated' ... ? Only if you choose an 1980's design, headless is the future, it is the opposite of dated. The 95% of basses that are essentialy 70 year old Fender/Gibson designs are 'dated' =)