Lightroom versio 6 ei käynnisty

Two thoughts-

1) You may need to re-create a Preset for an Additional External Editor that references Elements-2023.

This article describes how it is important to 'Choose' the "Editor", and NOT "Photoshop Elements"

ELEMENTS as Edit-in Lightroom

2) If you have installed the Adobe Creative Cloud desktop App  for any reason, then your Lr6 (old!) will have been converted to the subscription format and cease to run after a trial period.

What version of windows are you using?

Are you saying that Lightroom 6 will not run at all or won't launch photoshop elements 2023 when you want to edit a photo?

As a note to others who read this, Camera Raw is not included in the photoshop elements 2023 install to keep the installer file size down and has to be downloaded/installed afterwards by going to Help>Install Camera Raw in the photoshop elements 2023 editor. to get Camera Raw 14.4. Previous photoshop elements versions did include camera raw with the initial install.

Lightroom only uses Adobe Camera Raw with an "Edit In Photoshop" command (perhaps Elements) so there is no relationship otherwise. 

Elements has its own version of Camera Raw so if you downloaded a version for Photoshop, it's not going to run. The correct version for Camera Raw would be installed with Elements. You cannot install a Photoshop-hosted version and expect it to work with any version of Elements. While they share the same name, they are not the same product.