The Buddha put high emphasize on Anicca and thus it shouldn't be dismissed.
The Buddha put an emphasis on many dhammas.
The Buddha did not actually teach to constantly remind oneself of anicca because this would mean the mind is always thinking. Instead, one only reminds oneself of anicca when necessary.
The goal of meditation is to stop thinking so the mind has concentration so it can develop calm and then see anicca directly (rather than reminding oneself about anicca).
Reminding oneself about 'anicca' is called 'anicca sanna'. Where as seeing anicca directly is called 'vipassana'. Reminding oneself about 'anicca' is not 'vipassana'
Now my question: When should one remind oneself of the transience of all phenomena? I doubt in every occasion.
'Reminding oneself' is called 'mindfulness'.
The wisdom (panna) used with reminding oneself is called 'sampajanna' ('situational wisdom').
'Reminding onself about impermanence' is called 'anicca sanna'.
'Anicca sanna' is a preliminary practice (mentioned in MN 62 & MN 118) & particularly useful when there is a very strong hindrance that one cannot control and thus must endured; patiently waiting for the energy of the hindrance to dissipate & dissolve.
It is also done whenever necessary when something must be endured or when something is lost. Such as when you dislike a poster on the internet or lose a debate on the internet. You can think: "This internet is impermanent; this debate is impermanence".
For example, you have a good meditation, which then ceases, due to hindrances. If the mind clings to the good meditation wishing to have it all the time, one can reflect on impermanence to get rid of the craving.
Or if your mother nags you to do your chores, you reflect: "This nagging is impermanent; the sound is impermanent; doing the chores is impermanent".
If I for example walk or do my daily chores, according to the Buddha I apply Sati on the sense that I remember to not to forget walking.
No. The purpose of sati is not to remember you are walking. The purpose of sati is to remember to walk with wisdom or to do chores with wisdom.
For example, in a monastery, everyone must do chores. The chores are done with the wisdom that defilements lead to suffering therefore to do chores without defilements.
So this then is a rather concentration excercise without pañña (wisdom), right?
Sati is never without wisdom. Sati always has sampajanna. Sati is only remembering or calling something to mind. Mindfulness must remember to apply wisdom. Therefore, mindfulness & wisdom always operate together. There cannot be mindfulness without wisdom. This is why the term 'sati-sampajanna' is often used, as a compound.
Why would we walk or do chores without wisdom? If we do not have any wisdom about the purpose of walking meditation or doing chores & how to do it, why would we do it? For what purpose? With what method? For what benefit?
This video might help: Buddhist Meditation Sati Sampajanna
This book might help: Part II. The Use of Dhamma