Here is what I found in the last couple of days:
I already tried the code served from a Windows machine as well. The issue persists. I also figured that the indicators update rate AT THE TIME WHEN THE PANEL IS LOADED INTO A BROWSER has the critical effect on the probability of the remote panel in the browser starting correctly (with indicators updating) or not. The larger the loop update rate the higher the probability of the panel to load disconnected from the VI. I just increased the default poll interval to 300 ms (from 100 ms I had before) and it significantly increased the chances to see a "live" panel upon loading. Interesting, but once the panel is loaded and running OK, I can decrease the polling interval even to 1 ms - it will be working! However, if I hit the Reload/Refresh button in the browser while having this high update rate, the panel will be loaded "dead" (disconnected) with almost 100% probability. Moreover, then there will be no way to get it back in a running (updating indicators) condition - because the VI in the background is updating the indicators so fast.
Naturally, I observed different probabilities of a "dead" panel loaded on the same machine versus a machine across the network from the server even with the same update interval. So, again, it seems like it's all about the ratio between how long it takes to load the remote panel into the browser and the interval between indicator updates the VI has at the time when it is being loaded into the webpage.
So, when you try my example, if everything looks fine, try to decrease the default value of the poll interval in the handler.vit. I am sure that, say, if you have it at 10 ms, you will get a "dead" panel on load with almost a 100% chance even on a fast machine.
The complete "fix" is to not start updating indicators in the VI right away but rather have the user to push some button on the panel to start the polling (this will guarantee polling won't start before the panel is loaded). This is just a work around, not a real solution though - why do I have to make a user to do this extra step? They already have to click on the panel to get rid of the "Control granted" yelow box. Now I have to ask them to click on some "Start Polling" button in addition. They won't like it. The want the page to load with indicators being updated right away.