Broken thermostat or am I missing something? When does thermostat trigger boiler to make heat?

That looks like an early Chrontherm stat? The blue was the temperature during setback, the red was the normal operating temperature. Slide them both past 65 or so. Does it kick on?

Is there a clock dial on the left or inside with pins to set the schedule?

The spring is a bimetal thermometer. Some of them have an adjustment to dial it in to actual room temperature.

You are correct, it is not connected to the functionality of the stat. But it would be nice if it was accurate. Like a clock.

There should be a clock with pins to place and make setback times to save fuel. The Blue indicator is the setback temperature, and the Red indicator is the "comfort" temperature. If you push the Red and Blue indicators together, it overrides the setback.

If you take the cover off completely (both the cover and the door you have lifted in that photo) you will have two mercury bulb switches.  Those switches are attached to a bimetal spring that changes its coil based on the temperature.  (Just like the thermometer you say does not do anything).  You can set the position of the mercury switches with the blue or red lever.  By setting the blue lever to 65° you will see that the mercury bulb will rock to the left as the temperature drops below 65° and rocks to the left when the temperature rises above 65°.  There is about a 1-1/2° to 2° differential on that swing from left to right and back again.

Item 14 are the mercury bulbs on the bimetal coil.  The way a bimetal coil works is there are two dissimilar metals bonded together and spun into a coil. As the temperature changes one of the metals expands at a different rate than the other metal.  This causes the spring to change shape based on temperature changes.  The coil will get larger or smaller on a rotating motion.  That rotating motion will change the angle of the mercury switch. 

There is also a heat anticipator adjustment that will provide a small amount of “FAKE” heat inside that thermostat to anticipate that the temperature may rise above 65° a little too far on some gravity heaters, and make the room 67 or more after the burner shuts off.  That is adjustable based on the type of heating system you connect the thermostat to. I is number 12 on the photo and has a more detailed illustration (Fig. 8) just below it. On that same note, The onboard clock also generates a little heat and the calibration of the thermostat takes that into account.   So if the clock motor is no longer operational, then your calibration will be off to the cold side by a few degrees. 

If the small red and blue pinwheel at the lower right of the clock (number 8 on the photo) is set on blue, then the blue lever is what the thermostat is set on.   When the clock spins and you have one of the blue or red pins inserted at a given time, the small pinwheel will change to the Red lever to control the temperature.  As each hour passes the clock may change from the red lever setting to the blue lever setting based on where the pins are set on the 24 hour wheel.  Since you have no pins in the clock wheel, the thermostat will never change from blue to red or from red to blue.   By placing both the red and blue levers together. You are basically eliminating any set back because both levers will be set at the same temperature 

I hope this helps in understanding your thermostat.

As long as you don't need WiFi control of your system, and you can compensate for the accuracy of the calibration, you have a superior thermostat for a boiler like yours. Don't get rid of it!

I disagree. No newer thermostat will be as reliable or as last as long. So long as you don't break the mercury bulbs there is no hazard. None. Zip. Nada.

Stop scaring people, @HomerJSmith , or if you must pick something real.

Edit: cone to think of it, @CoachBoilermaker , if you're really terrified of it, I'll take it.

The common logic is that for every degree you can lower the temperature in your home your heat loss will decrease by 3%. So when you set your thermostat to 70° for comfort, changing it to 69° will yield you a 3% lower gas bill. That is what was advertised in the 1970s anyway. Now by setting your thermostat to 68° because President Nixon told you to, you would save out country from being dependent on OPEC and reduce your oil or gas bill by 6%. If you could turn your thermostat down 5° lower for half the day, (62° at night and 62° when away from home) you could save an additional 7.5% (5 degrees x 3% got 1/2 of the day = 7.5%). So the Chronotherm was reintroduced with a battery back up so the clock didn't loose time.

then there were the folks that said this was all a government hoax and you lost all your savings when you turned up the thermostat to recover from the lower temperature. I have looked at this and did some unscientific tests on my own home back then and came up with this:

If you lower the thermostat 5° for 3 hours then raised the temperature back up to the comfort temperature, you will probably not save that much. But any lowering of the thermostat more than 3 hours will result in a savings. To set back the thermostat 5° for 8 hours, there wil be 5 hours of actual savings. So when you go to work for 8 to 9 hours (including travel) you will save 2% When you set back 8 hours while sleeping you will save another 2%. That is a total of 4% if you remember to do it. and that is the reason for the setback thermostat.

In the 2000s we called it a programable thermostat. Today we call it a smart thermostat

Not saying the modern digitals aren't good. They are, and have a lot of features people really like — and particularly with faster responding heating systems such as forced air or many heat pumps they really can save you energy. Maybe even enough to pay for them.

They are reliable, too — in the sense that while they are working, the do work. However, I doubt we will see too many pushing much past the 20 year mark…

True enough, @HomerJSmith . Be an interesting 15 minutes, though…