I don't know if an actual connection would. But as mentioned, there is a dedicated output for exactly what he wants to do anyway.
The only reason to connect directly to the limit switches at this point, would be to add the benefit of knowing when the gate is in the full open position; as opposed to just knowing it isn't closed. Something that might not even be able to take advantage of in the C4 interface, so possibly a moot point.

Ernie,
You've 'known' me for almost a decade now.
While no one is as anal as you, I probably come the closest to you.
The reason I didn't provide a model number, is because there isn't one printed on the board. [Link: photos.app.goo.gl]
As this situation evolved, and vendors started getting back to me, I was able to spend more time on the interwebs and figure out (by comparing pictures) what the most likely part number might be.
With that, I was able to download several manuals and get board layout to compare.
This started as a simple switch replacement request, but has grown into a much better solution of using already defined contact points as MFG intended.

That was the first thing I tried.
Somehow, and I still don't know why, it didn't work.
While using common (middle) tab and unused one, the board would go trouble.
Going inline with the signal would do the same. That one I expected actually.

Since then, I was able to figure out the board's model and found that there are dedicated contact points for what I'm trying to do, so I'll be pursuing that avenue.

Thanks for suggestion.

Brad, thank you for that detailed response and suggestions given.
I would love to use the open/close limit switch, but couldn't get them to work in the little time I had. Might be a simple diode issue, but at this point I'm ready to move on.

Yes, the C4 would need 2 contacts with the first solution.
Luckily, the C4-Z2IO gizmo is field configurable for 2 relays, 1 relay & 2 contacts, or 4 contacts, all the while having internal and external temp sensor and even humidity! Oh, let me not forget 2 IR outputs.
What's even better is that it can accept onboard voltages from 5 to 24 volts in both DC and A/C. It also comes with wall wart for standard 120VAC.
This makes it virtually universally compatible installation from security and access control systems, to HVAC and industrial equipment.

This is the option I'm going to go with.
Looking at the J6 alarm output, I can't find any info to suggest if the alarm is active while the gate is moving, opening or closing only or on while the gate is open.
There are wires already connected to that jumper, and it looks like they might be going to a speaker (siren), but I don't remember hearing any beeping sound while the gate was in motion. It's possible the speaker went bad, or someone cut one or both leads off to silence it.

The other option as you mention is the J2 connector.
Few I was able to track down the part number OOMNIEXB.
"Elite O-OMNI-EXB Omni Option Board" comes up with this one page "manual": [Link: afence.com]

Looking at it, I thought this was a simple sales sheet, but to my surprise, I can't find anything more detailed about it.
No voltage information, no description of when this is triggered, what current it'll accept or pass.

Pins 10&11 are for Burglar Alarm Output. OK. Now what? Is this a relay?
- Is it active while in operation?
- Is it active (open or close) when open?
- Does it timeout (timer) or is it shunt?
- Is it N.O. or N.C. (power failure status)?

At 50 bucks, I'll get one in and play with it.

Lastly, there is also an option for a Q400MAU, which is an external MAGLOCK RELAY.
Again, no info on when it's active, etc.
Obviously it's engaged when gate is closed, but does it shut off and stays off when gate is opening?
How about closing? Does it come on when limit switch indicates fully closed position or when closing sequence start (pull in assist/retain)?

We're getting more and more away from technical specs and I don't like it.
No wonder Radio Shack closed. No one knows what breadboard, resistors and capacitors are anymore.

why not add a couple of security mag sensors to it to get a nc/no , go no go status... easy peazy

Limited space in mounting is one reason.
The other reason is that I didn't install the gate or the operator and if/when the gate company comes for any service calls, I didn't want them to see anything they can/could mess with.