Your default template group index can catch these URLs and embed the relevant template you want to display to list products.
But I'd be cautious on the amount of effort it will involve for little benefit and more complexity elsewhere - not to mention the usability from visitors.
Shallow/flat "directory" structure is good (source: numerous web articles and numerous SEO agency advice), but shouldn't be to the sacrifice of logical structure and usability. But compare that with loading keywords into the URL, which is more beneficial for SEO.
Also balance that the parent gains more weight from it's child pages - so "products" SEO value is diluted by having alternatives like "new-products", whereas "/products/new" maintains that value and is logical from build perspective and visitors.
SEO Investment would be far better placed in unique content text for the listing landing pages and what SEO agencies call "hub" pages. Don't forget to remove any unique content for paginated pages - better still ensure that the paginated links are no-follow and no-index. So based on this, remember that your products shouldn't be indexed on the listing page - as that's not what you want the Google index to point at, but the product page itself, so a short URL for the listing is irrelevant (in SEO terms) for the product itself, just the landing/hub page (the same as any other content page).
In all ecommerce builds I've worked on, they've all been heavily category based for listings, so "new" is just another category and translates/identifies (seg2cat) to the correct channel:entries output.
Lastly, think about the duplicate development code (although you can use embeds to create a single source) and how difficult that would be to maintain, track changes and for others/client to maintain.
Good answers and article links: