Question

So that the Notion word count will always show on the page as one of the properties, without the need to check with the dots in the top right corner
Does any formula have this effect?

Answer 1

It is not possible.

Notion formulas work inside a table and word count work by page, so it is not possible to use a formula to count characters outside the table.

Hopefully that makes sense.

Answer 2

The formula to make it work in a database is possible with Properties but not text on the page.
length(prop("thePropYouWant") will give you a letter count for a Property. But if you want to count words, you can count the spaces. For example :

if(length(prop("Name")) > 0, length(replaceAll(prop("Name"), "[^ ]", "")) + 1, 0)

which uses every occurence of space