let's say you have a site loaded with pages with 2k+ word articles.

This describes my  site.

Would a lightly-populated index page (less LA Times, more Big Background) have a negative impact, beyond the obvious missed opportunity for having that page have more content as well?

If I reduced my homepage to minimalist content I would miss at minimum, 50,000 visitors per month.  The diverse words on that homepage pull in nearly 1/2 of its traffic and they also enable it to rank better in many SERP.

This homepage also holds lots of #2 rankings, where one of my article pages holds #1.  That occurs because the homepage is made relevant by the diverse words.

We can assume the minimalist index page would have appropriate page optimization, including some content (just not lots).

To me this means that you will get "some" traffic rather than "lots".

Many websites have more search traffic entering through their homepage than any other page on the site.  And it is usually the strongest page to use in battle for difficult keywords.

We seem to be from different perspectives.  I sense that you are married to a visual effect... and I am married to using my homepage as a traffic-pulling machine.

There are design features that allow a page to contain a lot of spiderable text that is revealed with a click and pages that rotate visible content one cell at a time but hold lots of total text.  Perhaps one of those would be a way for you to have both.

are there any SEO downsides to not having any substantive content on the home page

These are not thoughts.  They are facts.  If you have a homepage with just a few words you will get a certain level of traffic.  However, for almost every relevant word that you add to the page you will probably get more traffic as those new words combine with existing words to create many more combinations of new queries for which you are relevant.

Here is a short story....

We had a lot of pages that had one sentence descriptions of photos.  They got a little traffic.  When we increased the text content on those pages to a couple hundred words the rankings increased and the traffic went up by 10x.... When we upgraded those pages to 2000 word articles with several photos the rankings went up again and the number of visitors went up significantly - some to thousands of visitors per month.  Now some of these pages get more visitors in ten minutes than the one-sentence pages received in a day.

Our guess is that if the home page doesn't have much copy, that odds are that other specific pages will tend to perform better for non-brand search terms, which seems OK.

I think that your guessing is harmful to your wallet

If people DO find the homepage, it would likely be a brand search or an ad referral, in which case the minimalist, non-copy design would be conversion-friendly. Does that theory hold any water?

Most of the visitors who enter my site by the homepage are coming from queries that have nothing to do with my brand.  My homepage looks closer to the LA Times than to what you are describing.  I want my visitors to say DAMN!  Look at all of this stuff!

The retail sites where I have this convert really well.  I would be very hesitant to use your proposed homepage unless I was selling just one item on that website.

One of the easiest ways for you to rank higher than those sites would be to setup author profiles and claim yourself as the author of the original content.

I believe that this is a very good idea.  However, I am confident that it will not solve the problem.  Google is not giving strong weight to claimed authorship at this time.

Your strongest weapon is DMCA actions and to stop giving others permission to use your content.

Your site could be suffering from Panda ranking reductions as a result of your own content being wild on the web.