There are a number of reasons to have a wall mounted thermostat.
1) To stop system freezing.
2) To auto switch off boiler in summer months.
3) To allow remote control.
4) To comply with silly laws.
5) To control temperature of the room.
I can't see from picture type of boiler, but I can see the thermostat is rather basic, so could be froststat, but not a modulating thermostat just simple off/on, and today most gas boilers modulate, (turn up/down) so the control needs to be a modulating control, most common is a thermostatic radiator valve (TRV) however the TRV has a problem, it can't tell a boiler when to run, in theory you could design the boiler so once it hits lowest output and the return water is over temperature it switches off, but what you can't do is once switches off, know when to switch on again.
So you want a thermostat which will turn off the boiler in summer, and turn it back on in winter, the difference between off and on can be large, so missing out the neutral works well.
So the scenario, the TRV controls bedroom heat say 18 degs C, the thermostat is set to 20 degs C off, 17 degs C on, so all winter it does nothing, when we get a warm day in spring it turns off heating and when we get a cool day in autumn it turns back on again.
I have a similar system, OK because the thermostat is so high it may need to be a little higher temperature, but once set what every room it will never need touching again.
OK a Nest Gen 3 thermostat is clearly designed to connect to the boiler ebus using OpenTherm protocol and regulate the boiler output from some central room, but that thermostat is a very simple cheap one. Of course you can take it a little further, a programmable wall thermostat and programmable TRV head can be set to work together and allow for it getting cooler over night, but today Nest Gen 3 is rather the odd one out, most of the higher range thermostats talk to the TRV head.
We need to remember it is the TRV which controls room temperature in most cases, not the wall thermostat, there may be the odd open plan house where a single thermostat on the wall will work, but in most cases today we need multi thermostats one or two for each room, even if only the cheap eQ-3 with bluetooth so a pair can talk to each other when controlling the same room, or without bluetooth if only one radiator in the room, it is the TRV head that does the controlling in most cases today.
Wall thermostat only turns off system in summer.