Hello there,
if your SEAT has done 80k with the original glow plugs still fitted, it'll more than likely be the issue, plus it wouldn't hurt replacing the Thermo Transmitter also known as the coolant temperature switch, as one of the guys has already mentioned.
The ECU won't always log a fault code for a failing sensor, it depends how well it's still working, and whether it's an intermittent fault.
The coolant temperature switch you can test them with a multimeter, at zero 0°C you should see 7000 to 8000 ohms, then at running temperature 85°C to 90°C you should have a reading between 300ohms & 200ohms Basically the resistance should drop down the higher the coolant temperature gets.
From watching your YouTube video the engine struggles to start up, but does eventually get through the rough lumpy running, including the diesel knock that seems to disappear if it didn't then I'd say you have a faulty injector/s, but it does disappear.
White smoke is neat diesel (unburnt diesel) coming out of exhaust, faulty injector/s plus is often accompanied by diesel knock.
On rare occasions it can be coolant being burnt from a cracked cylinder liner, or defective cylinder head or head gasket as mentioned already by a few others.
If it's neat unburnt diesel it'll really smell strong and will linger around in the air, coolant doesn't as it's mostly steam coming out.
Black smoke is over fueling or faulty/blocked EGR valve, or faulty diesel particulate filter.
Hints of blue smoke are oil being burnt in cylinders, whether that's from valve stem oil seals or piston rings, or badly warn turbo bearings / seals.
You can add some fuel additive/cleaner into fuel tank as already mentioned by others, it wouldn't hurt to be honest.
But I still think it's your glow plugs that have seen better days, this happened to my brother's Peugeot 206 a few years back, only had 2-1/2 glow plugs working, put a new set in and it started up first time.
The old glow plugs you can test them by simply using a power probe, or by using a foot long piece of wire (use 1mm to 1.5mm thick wire)
TEST GLOW PLUGS
Put the body (the threads) of glow plug onto the negative (earth) terminal of a battery, then attach the wire to the back/top end of glow plug, other end of wire goes onto the positive (Live) terminal of battery.
The glow plugs should take between 3 to 5 seconds to glow red on the heater tips,
if they take up to 8 seconds to heat up or don't heat up at all (glow red) then the plugs are no good and need replacement with new.
Hope this helps
Si