I create a lot of Captivate projects from PowerPoint presentations and load them in my LMS. I was informed by a peer that they will no longer work once flash goes away. Is that correct? I was instructed to run the HTML5 Tracked, which I have done. They all have the error message "Only animation image is supported". Everything runs just fine when I publish and run the in the LMS, but will this be a problem down the road? Or can I just leave them as is?
Just publishing and running in the LMS is not a good guide unless you are publishing ONLY to HTML5 and testing in the LMS. You may be publishing to SWF and your default browser may not currently have a problem with playing SWF. But that's not going to be the case next year.
This screenshot seems to indicate you have a lot of animations in the project file that are not supported in HTML5. My guess would be that these are SWF Flash animations. Flash is not supported by Adobe after this year (2020) and is already unsupported by many web browsers. You should not be thinking of just leaving this as is because your content is not going to work on a browser that does not support Flash.
You can quickly see if your content is going to work or not by just publishing ONLY to HTML5 and then try to run it in a web browser (e.g. MS Edge or Chrome). Any Flash SWF components will simply not appear.
Be aware that the Publish dialog is a little confusing at first and you need to be careful when selecting the options so that you ONLY publish to HTML5. There are two buttons side by side on the dialog. One says SWF and the other says HTML5. You need the HTML5 button ONLY to look "pushed in". Like this:
If you publish your output this way and then run it from a web server environment you will very quickly see whether or not you are going to have problems.