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I couldn't find the motor searching on the part number "Dayton CHM..." but found a similar Dayton gearmotor which retails for about $100 (yikes).  Anyway, it's a sealed gearbox so ignore anyone who asks you to open that up to look for backlash or meshing issues!

0.45 RPM output is about ~3 degrees per second and it looks like about that speed in the video so I'm thinking the motor is working fine, there's enough voltage, etc..  As you point out, you can hear the gearmotor spinning when reversing direction several seconds before the TT moves.  So I'm thinking the gearmotor output is 1:1 ratio or directly driving the TT platform.

Does the TT "lift off" the motor "D" shaft?  This would be at the, say, yellow arrow in photo above which I copied off your video.  I assume when you said 1/2 shaft in your original post that this is what you meant.  So let's say the D-shaft is as shown in black.  Then there's a coupler/sleeve on the TT depicted in red.  Obviously this needs to be a glove fit over the D shaft.

Let's say the sleeve is NOT an exact glove fit and there's a gap as shown in the lower red sleeve.  This would exhibit the behavior shown in video.  That is, there would be "play" in the TT of several degrees.  When changing direction, the flat-face of the black D-shaft would have to rotate a few degrees to re-engage the flat-surface of the red sleeve.  If stopping and restarting in the SAME direction, there would be no delay since the flat-face of the black D-shaft is already in contact with the flat-surface of the red sleeve.

A set-screw(s) depicted in blue can help to securely mate the two parts so there's no wiggle room.

That's my 2-cents. 

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The amount of play you are showing in the bridge at the top of this page is way more than what Stan is proposing is possible from a shaft flat in my opinion.   

I suspect you have a "clamp" collar or set screw loose somewhere on the main drive shaft.  I am kind of guessing that there is a "shaft coupler" somewhere that is allowing that much rotational play...   You should check to determine the following:   

 when you move the turntable bridge by hand as you did in the video is the shaft coming out of the drive gear box also rotating ???   you may need some help so that one person can be at the drive while the other moves the TT bridge. 

That looks just like the motor and gear box on my Millhouse turntables.  I use a MRC 6200 and Dallee Yardmaster dc power packs to drive my Millhouse TTs.

Using these better power packs  the speed can be easily controlled and finally feathered for a perfect visual alignment every time.  You vision and distance to the TT  can be an issue.  Most all of my TTs are very close to an aisle.

Rock the TT bridge through it's full loose arc then back it off half of the arc which centers the set screw on the midpoint of the flat spot. then drive the set screw in firmly.

I would suggest to NOT used fixed voltage applied with an on-off switch to operate the bridge.  Consider using a variable throttle  of a quality higher amp power pack to obtain excellent control.

Eternity is a long time to have been wrong.

I figure you'll resolve the "play" issue (please let us know).  As for the final alignment, to repeat my previous musings, I find it hard to believe you can find an off-the-shelf or plug-and-play optical (or whatever) module that performs the required function.

That said, if wiring, soldering, assembly, etc. is not in your comfort zone I was thinking about those super-miniature train cameras - they fit in N-scale for example.  They are down to about $20 or so.  They can send a "live" picture via Wi-Fi to whatever (whatever being an old unused smartphone, tablet, etc.).  This would take some experimentation with focusing and whatnot, but perhaps you could mount the camera on the rotating TT aimed at the incoming tracks.  The camera most likely requires DC power which would be derived from AC track voltage already on the TT using a $2-3 AC-to-DC voltage converter module.

I was finally able to get back to this.   I was dreading pulling the table back out because everything had to be undone.  But it’s done. 

 

The motor is not direct drive to the shaft. I was wrong.  It attaches to some kind of gear box.   I found that there are two set screws.  None of the Allen wrenches I had on hand matched either screw.  As per prior posters advice I will try to tighten them as soon as I remember to take a full set down to the layout room. 

 

In the meantime here are pictures of the underside. CA5FB7E4-6C6B-4196-88A2-A55CD30F5C1C7166E732-3248-4FDB-B245-87FAC62FCEB1CE90598A-0068-4E1F-8940-2F71ED82CA40

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gear d shaft

The motor and gearbox come together as an integral unit as shown on the left which is from the same manufacturer.  I'm interested in those 2 set-screws.  I'd think they would go on the top side and into the flat surface of the D-shaft.  Did you take photos of that?  I can't imagine why a builder would have used an odd-ball (or metric) screw size for something like this...in other words any standard key set ought to do the trick.

sae key set

 

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I'm getting the "Image Not Found" blank white square for what might be your 1st photo.  Presumably this shows the upper set screw?

I now see the lower set screw in your 2nd photo...recopied here for the convenience of anyone else following along!

set screw bottom

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