Image Processor not applying Color profiles

As an aside, the OP and perhaps others should start a feature request* that IF Convert to sRGB check box is on, the 2nd check box (which IMHO should just be removed; Include ICC profile) also be checked.  

There's no reason to have the current behavior which is just a big hurt me button (convert, yet embed which is again IMHO, incorrectly labeled "include"). Any opportunity for Photoshop to allow untagged RGB mystery meat needs to be removed. 

*To do so:

https://community.adobe.com/t5/photoshop-ecosystem-ideas/how-do-i-write-a-feature-request/idi-p/12384447

Also on a Mac, the latest version of PS: With the embed check box invoked, the images do get tagged with sRGB (and converted from other RGB Working Spaces). 

If you don't want to start from scratch, just Assign sRGB to those images; they are in sRGB. In the future, just use that checkbox, it does work for me on Mac. 

I am testing on Windows using Photoshop 23.4.2 and everything works fine. I am running Image Processor on one image with Untagged and two images in Adobe RGB (1998).

Processed images are all converted/saved to sRGB profile.

The files are untagged because in the action is not required to embed the current profile.  Re-Record the action and include the check box to embed the color profile as shown in the screenshot below.  That will resolve this issue.