Camera RAW plug-in not plugged in to LR Classic.

Here is my 'take' on your problems-

I cannot import Canon RAW CR3 files into LR Classic

Why? What error message do you see?

Do you have any 'Permissions' problems?

Is the destination folder available with full Read/Write permissions?

Is there space on the hard-drive? Is an external Hard-drive (destination) available?

Is a Card Reader, usb cable,  not working?

because the Camera RAW plug-in appears not to be installed although it appears to be installed on the adobe panel (see attatchment) and manual installation does not work.

Lightroom-Classic does not need the ACR Plugin. LrC works independently without the plugin.

It does not apear in my plug-in list in LR.

It does not need to. ACR is automatically installed for use with Photoshop.

I have no adobe profiles visible nor the recently added Canon Profiles.

Not all cameras have profiles supplied! See this forum link for info-

https://community.adobe.com/t5/camera-raw-discussions/p-camera-matching-profiles-requests-and-information/td-p/12354741

Official Adobe stance on this:
” Support for a particular flavor of a camera’s raw file does not necessarily mean that there will be Camera Matching profiles for that camera at the time of raw file support. It is normal for Camera Matching profiles to follow raw support by one or two releases. In the case of some camera manufacturers or the event of major changes by a particular manufacturer, that time may be greatly increased.

Note: Not all cameras will ever receive a set of Camera Matching profiles. ”

I am presently converting my CR3 to dng externally and importing as such. This is not satisfactory.

You do not need to convert to DNG. the Canon R6 has been supported by Lightroom-Classic since Version 9.4!

Your CR3 files should IMPORT directly into the LrC library AS CR3, by using the Import dialog in LrC v11.4.1

because on the 'Open with' dropdown neither LR Classic nor PS appeared.

This is because LrC is not designed to use "Open With" from File Explorer to 'Open' photo files in LrC!

Lightroom-Classic will only 'Open'  CATALOG files with the file-type suffix of .LRCAT.

Only Photoshop will 'Open' photo files by the OS 'Open with', (and raw files will first be rendered by ACR).

You need to IMPORT image files into the LrC Catalog, then use LrC to work and develop.

You should NOT be seeing LrC in the 'Open with' drop-down context menu!

You might see Lightroom (Cloud/desktop version) in the 'Open with', but again this only IMPORTS an image to the Lr catalog.

I hope my comments are helpful.,

You stated, " I am presently converting my CR3 to dng externally and importing as such. This is not satisfactory. ".

Could you clarify, are you using the Adobe DNG converter, if so what version?

Also supply a screen capture showing the options you have selected for the conversion.

In that second screenshot, showing the profile browser, was the image selected a CR3, or a DNG?

The Camera Raw plugin shown in your screenshot is for use with Photoshop and Adobe Bridge. Lightroom (Classic and Desktop) have a version that's fully integrated (i.e. it cannot be added separately).