Your requirements seem quite site-specific. If you have any development resources there who work with C++ or C#, perhaps you'd consider having them write a
plugin to create the visualization you're describing? If you don't have internal resources (or friends/family who code) I will encourage you to participate in the FOSS model by helping to support development of features your company values. Part of my business is to provide ad-hoc development services in C# and I would welcome an opportunity to create a solution for you which we can then publish as FOSS for others to use. Perhaps you could offer Dan the same opportunity. We all use a lot of freeware and developers easily get burned out if there is no motivation other than joy to support it. Businesses spend a lot of money on commercial software and they also appreciate the benefits of FOSS. It's appropriate for users, IT Management, and business owners to recognize the equity in helping to support the model by supporting the offerings that they value.
For free options within the limits of the existing application ...
I only use Gantt when I have a schedule, but in most cases my projects are just priorities and related tasks. I'd rather not use the built-in fields for something other than their actual purpose. The same prioritization can be accomplished if we could use Custom Attributes with date fields, and an alternate Gantt view based on Those fields.
It occurs to me that an alternative to this pointer mechanism is to use numbers for steps, like a BOM system. For example, the morning schedule:
100 Get up
200 Shower
300 Shave
Oops, don't forget the teeth!
100 Get up
150 Brush teeth
200 Shower
300 Shave
Well, I prefer to shave before showering:
100 Get up
150 Brush teeth
180 Shave
200 Shower
400 Get dressed ...
So that's yet another text-based sorting mechanism, essentially a different priority system. Just use the normal column Sort feature to see tasks in order, and maybe filter out tasks that don't have a sequence number. Have a Lot of tasks? Use sequence numbers in the range of 1000000 to 9000000 rather than 100 to 1000.
As far as left-side, right-side, just drag the column to wherever you want to see it. :)
But to visualize that in a way that pops out at you, you'll need someone who can write some kind of code, any language, doesn't matter. Create a UDT that gets the sequences and assigns a color or icon to tasks. It should be easy to translate the increasing sequence number across a gradient from red to green (lower numbers get done first and are therefore red=hot). You can also save your current position in the sequence, so with the example above, if I'm shaving, the next hot task (pun intended I guess) is to shower, not to get up. All that is within the UI itself.
Now consider that we can also export and transform tasks, and then manipulate the data with XSLT. XSL can convert the XML data to HTML and of course with HTML we can do all sorts of visualizations. You can simply load the data into a JSON definition and let JavaScript do some amazing (and dynamic) task re-ordering and rendering. I think a combination of the above and this will accomplish your goals (all free and with local resources if you have them).
As an alternative to HTML/JavaScript you could transform the data into pure data, CSV, JSON, or XML, and then open that with another graphical reporting tool that renders the data exactly as you wish. This is what happens when we use the Export feature - and with the plugin mechanism described above you can create your own export target.
Whenever I save my tasklist (automatic every 2 minutes) it does a transform, exporting the data into HTML with a nice structure, coloring, etc, and that then gets published to my website. With this my clients can see to-the-minute what I'm working on and how much time has been spent. Picture that same mechanism being used where you just need to click a toolbar icon to display that web page.
That scenario gives you Exactly what you want with no changes to this software whatsoever.
If you want a built-in site-specific solution, you may never get it. If you then do not make use of development resources, you will certainly never get it. But if you can motivate people to do what you want, you Can get the business functionality you want, though probably not as it's been envisoned. At this point it's not a matter of possibilities, but choice and priorities.
HTH
T