On each slide, can I hide the playbar until the slide timeline finishes?

If you execute this JavaScript on Slide Enter, it will do what you want:

var csTo;
window.cpCmndShowPlaybar = 0;

window.cpAPIEventEmitter.addEventListener( 'CPAPI_SLIDEENTER', function ( e )
{
csTo = e.Data.to
window.cpAPIEventEmitter.addEventListener( 'CPAPI_VARIABLEVALUECHANGED', runSlideTimer, 'cpInfoCurrentFrame' )
});

function runSlideTimer( e )
{  
if ( window.cpInfoCurrentFrame > csTo - 5 )
{
window.cpCmndShowPlaybar = 1;
}
}

You can also create a progress variable and it will show the playbar if you have completed the slide.

Another possible solution would be to add a little script to your onEnter action.

setTimeout(function() {
cpCmndShowPlaybar=0;
},250);

setTimeout(function() {
cpCmndShowPlaybar=1;
},2500);

setTimeout(function() {
cpCmndPause=1;
},2990);

I will explain the above code a little. This would be placed on each slide as an onEnter action with some slight modification depending on the overall legth of the slide.

We see three setTimeout blocks above.

The first one will hide the playbar after a quarter second (250).

The second one will show the playbar again after 2.5 seconds (2500).

The final one would pause the slide after 2.99 seconds (2990).

You would want to modify the time on the second and third blocks. Modify the second block to be about a half second shy of the full length of the slide. The value is in milliseconds so 1000 is equal to one second.

Modify the final block number value to be just a hair short of the full slide length. In my example, the slide length is 3 seconds (3000) so I set the pause to happen at 2990.

The pause is so that the learner would use the playbar to advance if that is what you are after.

You can use an Advanced Action triggered ON Slide Enter to hide the Playbar, but you don't have the option of hiding just one component of the playbar (e.g. the Next button). 

Once hidden, the playbar will remain hidden until another action shows it again.  However, Captivate is a little limited in regards to options to trigger the SHOW action that do not involve the user doing something.  You do have an event at the end of the slide (called On Exit) and you could set up an Advanced Action to be triggered by this event that first Pauses the timeline (before it advances to the next slide) and then shows the playbar.  However, most seasoned Captivate developers will advise you against using this On Exit event because it actually fires right on or after the final frame of the slide.  Any split second in mistiming and the event may not fire in time and your action does not happen.

Another option is to add another couple of actions to the Advanced Action triggered On Slide Enter of the slide, this time using the Delay Next Action By action to specify how many seconds should elapse before then executing another action to Show the playbar.  However, the down side of this is that if your slides are all different lengths then you need to use a different Advanced Action for each slide.  A Shared Action might be worth considering in this situation because it gives you the option of using a parameter to specify how many seconds the delay period should be.