I came up with a work around for this, though there's one aspect I still need to address, if anyone has any thoughts. Under Tax & Calculation Rules, I created a new Tax rule for each shipping amount. For each rule I set the "Type of Arithmetic Operation" to Tax per product and then set the value to the amount I wanted that rule for; and gave an descriptive name for what it is - the tax rule for shipping rate of $6 was called $6 Shipping, the $10 one was $10 shipping and so on; you can use whatever convention you want. You then go in to each product and from the drop down menu next to base price you select the rule you want applied to each product for shipping.
The next step is a hack of the code.
In /components/com_virtuemart/language/en-GB/en-GB.com_virtuemart.ini
Find
COM_VIRTUEMART_CART_SUBTOTAL_TAX_AMOUNT="Tax"
Change only the part in quotation marks ("Tax") whatever you want the column to say, "Shipping" or "S&H", etc. Make sure whatever you want to change the text to appears inside the quotes.
Also find
COM_VIRTUEMART_PRODUCT_TAX_AMOUNT="Tax amount: "
Edit the text in the quotes here as well. Where the first one is the column that appears in the shopping cart, this one appears in certain instances, like as part of the short description when listed by category.
There are other lines in this file with labeling lines referring to tax that you may or may not need to change, depending on what you what turned on to appear in the cart; so you should do a word search for the word tax, and using use your discretion whether or not to change it. I may likely go back through and do that too, just to be on the safe side.
And of course, save changes and re-upload, or however you chose to edit those types of files.
Done this way, each product ends up showing it's own shipping total separately, which I'm okay with; though I am going to look further into at what point it will show the grand total of the "shipping" (formerly tax) amount, and the product prices together; which it hasn't done yet, despite the options to show that being enabled.
The final part to this I just need to resolve, is to "turn off" the original shipment method line(s); so that they are neither necessary or displayed. I don't want to just create a $0 shipping method, as that would be confusing, obviously. This is where, if anyone knows how to do that, it'd pretty much be set and ready to go.
I'm sure the weight based shipping method could be edited and adapted as an alternative way of applying tax, if that were needed in addition to flat shipping.
What would be even better would be to find a way of copying all the necessary code, et al, for the Tax & Calculation Rules, and find a way to rename it and incorporate the basic functions as a shipping method, adding that extra drop down box too, to each product, configuring shipping that way. Perhaps someone involved with the Virtuemart project would know where all of that would be, and how that would work; and ideally wouldn't be as huge of undertaking as it might be for others...