If you don't want to be a concert pianist, and want to, as you said, just play a few songs (to begin with anyway), then no, starting late isn't a problem.
There are a few reasons why beginning early is, however, better.
But it should be noted that a poor musical education from a young age can do great damage, and create huge errors in playing, so this whole issue depends on viewpoint.
Some benefits of learning young are:
More adaptable fingers.
This is because the fingers are bearing the strain of playing whilst actually growing. As such, the muscles/tendons/ligaments form accrding to this strain (just as someone who grows up in an area where they need to walk a lot will have developed leg muscles). The result of this is less strain on the hand when playing music, since the hand has already adapted to potential strain it may be placed under.
Familiarity with repertoire.
This is often overlooked, but learning young usually means developing good sight-reading skills. If one were to begin at a young age, say, 4, by the age of about 16, the player would probably have the ability to sight-read through the Mozart and Schubert sonatas in the space of a week. Good sight-reading is a great skill to have, not only for the sake of sight-reading, but for the sake of learning otherwise unknown and inaccessible music. If I see some sheet music I have never seen before, I buy it, simply for this reason.
It is amazing to think that in the space of about a year, a good player/reader has the ability to familiarise themselves with over 200 years' worth of repertoire.
There are many reasons for starting young - these are just a few.
There are also just as many reasons for starting older - the case can be argued both ways.
One benefit of starting older could be adding more emotion/interpretation to a piece that could otherwise be disregarded as yet another sight-reading exercise for the younger pianist.
I saw this once - a PhD student was performing the Beethoven bagatelles (technically not the hardest pieces in the world), and a few days later I heard a young 'virtuoso' prforming the same pieces.
The difference was worlds apart - the PhD student exuded soul and understanding, whereas the young guy sounded like nothing more than a mere mechanicus.
So it all depends on perspective.
Hope that helps a little.
PS if you do want to play, then play, and don't let age put you off.
