Hello PiotrMisiuna,
I had a look to the
TNY278 datasheet and
application note.
IMO, PI devised a really complicated chip :
- EN/UV (pin nr 1) has dual function
- BP/M pin (pin nr 2) has 3 simultaneous functions

Among 75 pages of documentation, the sole clue about an implementation without an optocoupler is
Figure 28.
Miele probably designed his own schematic around the TNY278 without an opto on the BOM : 20 Cents saved per board

This would be more than a day harship to fully understand how this chip works and what are every possible reasons that avoid the full start, or entering into protection or hiccup mode.
TNY produces internal 5v so I assume is good.
Not sure : internal regulator (pin 2) should provide 5,85 VDC and below 4,9 VDC it triggers shutdown.
If you mesure 5 V, it's weird.
Q1) Check again the voltage at pin 2
Q2) Try to mesure VBias (see
Figure 28)
Q3) Can you probe the drain (pin 4) with an isolated scope or differential high voltage probe to check if the IC hiccups ? (normal mode : switching around 130 kHz)
Q4) Did you check if the board behaves differently when inserted in the appliance ?
Q5) Is the UL339 (LM339) somewhat involved in monitoring/controlling the TNY278 power supply ?
(follow the tracks to see how it is interconnected)
Q6) Are there parts on the other side of the PCB ?
Q7) How many windings does the ferrite transformer have ?
Q8) Did you
ring it ?
I'm wondering if would not be easier to bypass the TNY278 power supply by using an external 12V SMPS.
Q9) Can you try to check if the appliance works when feeding the board with external 12 VDC ?
Using a lab power supply to perform the test would also allow to mesure the current drawn. Table 16 in the AN says 21,8 W max for the TNY278. A 2 A SMPS should be sufficient.
Some external 12 VDC 2 A SMPS :
https://www.reichelt.de/de/en/plug-in-switching-adaptor-24-w-12-v-2-0-a-hnp-24-120l6-p177030.htmlhttps://www.reichelt.de/de/en/switching-power-supply-25-2-w-2-1-a-12-v-snt-rs-25-12-p137085.html