Push-button starters aren't new.  Companies like Lincoln, Packard,
Hudson and Cadillac used them from the 1930's to the 1950's.  In fact,
the idea of having an ignition "key" wasn't around from the very
beginning - most cars prior to the 30's probably used push-button
starters.  I think most companies quit using them by the 1950's,
though.

As for new cars, they've only been using push-button starters for the
last few years.  The Honda S2000 has used one since 2000, and I can't
remember if there were any mass-produced cars that used a push-button
starter in the 1990's.

Also, there are two types of "push-button starters".  The one in the
Honda, for example, behaves just like a regular ignition switch.  You
have to press and hold the button until the engine starts.  Other
cars, like the Aston Martin, only require that you press the button
briefly.  After that, a computer takes over and continues to turn the
starter until the engine is running.

As for push-button transmissions, plenty of American cars had those
well into the 1960's.  The most-recent one I can think of off the top
of my head was a friend's 1964 or 1966 Corvair.

1912 Cadillac.  93 years ago.

Traditionally, the ignition switch and starter switch were separate.
The modern, four-position (acc/off/ign/start) key switch was introduced
as a "feature" in the early to mid-1950's, to save drivers from the
horrible inconvenience of having to turn on a key AND push a button or
a momentary toggle :)

I think on modern cars the pushbutton is there simply to give the
interior of the car a "racy" feel; most race cars use standard toggle
switches and pushbuttons for the electrical switches as reliability and
serviceability are more important there than aesthetics, and most race
drivers don't seem to be inconvenienced by having different switches.
Thus to give a street car a "race car" aura, the designers use a more
expensive, stylized version of the standard starter pushbutton.  Eh,
makes no sense to me, but whatever.

Incidentally, prior to most cars adopting key-start, aside from a
dash-mounted pushbutton or momentary toggle, there were also switches
mounted under the floor and activated by buttons looking a lot like a
floor mounted dimmer switch.  I know Studebaker and possibly others
used to mount it underneath the clutch pedal, and I think Buick had it
under the accelerator?

Finally, regarding another poster's comment about high current, I'm not
aware of any vehicle made in the last 50 years that didn't have a
starter solenoid; I doubt any more than 20A is flowing through the
"start" contacts on the ignition switch or starter button.  However, in
very, very old vehicles, you'd sometimes see on top of the starter
where you'd expect there to be a solenoid, instead a very heavy duty
switch.  For example, I was working on a '50 Studebaker Commander a
while back; the starter was on the driver's side of the engine and had
one of these HD switches on top of it, it was activated by a pull rod
that hung under the dashboard next to the OD handle.  Obviously a
clutch/neutral safety interlock is impossible with this setup which may
be one reason why it's dropped out of favor and solenoids became the
way to go.   I think that car had a manual choke as well, IIRC.  Three
things you never see anymore :)

nate

Many Jaguars had push button starters. My 1965 E Type did.

I believe most of the push button starters you are seeing on Lexus, Acura,
Mercedes, Infinitis, and BMW's now are because they have 'keyless'
ignitions.  (radio fob in pocket identifies to system that this is valid
operator). If you don't have a key, you need a way to tell the car to
'start'.  Although they may be 'sportier', I think the real answer is more
practical.

I also recall push button starters.  My first car, a TR-3, had one,  Then I
believe, as an earlier respondent suggested, a separate switch to hndle the
high current of the starter motor was probably cheaper than a more complex
and higher current ignition switch.