Is it correct that I should push 1 button for ten seconds and then they all start blinking together for like a minute, which indicates they should be pairable?
I have a Moes one button switch. Pairing was easy. Problem with zigbee2mqtt is that it only reports battery status. No action available. Pretty useless.
I also have only the battery in HA.
Removed and paired it multiple times from/in Zigbee2MQTT and even the battery from the switch.
I also restarted HA multiple times.
Zigbee2MQTT says its a ZT-SR-EU4.
MOES site says its a ZT-EU4-WH-SR.
Both you and @janneniemensivu need to enable the legacy action sensors from the Z2M config.
Those sensors were disabled by default since the beginning of this year, so newbies are mostly unaware that the action referenced in the docs is legacy stuff.
Alternatively, go with the recommended or experimental way of using MQTT device triggers or event triggers in order to future-proof your automations to a certain extent.