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In my opinion, no.  Moving to another platform puts you another iteration away from the original solution.  If you do move and things don't get better then what?  You'll have double the 301's to address.

My recommendation would be to restore your store back to Volusion to verify your conclusions but I suspect you 1) did not make a full backup of your store's data and data files; and 2) you did not keep your Volusion store active (no-indexed and not attached to your domain) until you were sure everything was in proper order.

Don't forget that the Volusion Product Name Short field serves two purposes: one, as the URL and two, as the anchor text on your category pages (which directly affects your internal link structure).  It could be due to your breadcrumbs as Volusion takes the primary category of your site and uses it as the breadcrumb.  I'm not sure how that got transferred over either.

I think your BC store is salvageable but not without fully investigating the issues via your store and your analytics...as well as your back link profile.  For all we know it could simply be the latest Google updates.

Curious why you didn't keep your current Volusion URLs?  With BigCommerce (which we are a partner), you can match the Volusion URL and forego the 301 redirect if you feel that's critical to your SEO success.  A proper 301 redirect should be adequate and not account for such a dramatic drop of traffic.

As an ecommerce consultant you need to understand where your traffic was coming from and make sure that's protected when you move.  For example, if your products have a lot of Facebook Likes, when you change the URL, you will lose 100% of those Likes...and this should be a factor in choosing a new platform.

You should be able to diagnose what's happening from your Google Analytics.  Take a look at where traffic has dropped.  What landing pages were driving traffic before and no longer are?  What keywords have disappeared and what pages were appearing for those keywords?  Are those pages 301'd correctly?  What products and categories were driving your traffic and revenue?

Obviously check all your pages for proper meta tags (INDEX, FOLLOW)...with these "simple" platforms, it is easy to accidentally make your whole store disappear.

It sounds to me like something was missed during the migration and you're now.  We are a top Volusion consulting firm and have worked on hundreds of stores...and we're partners of both BigCommerce and 3dcart.  You can have a successful store on a hosted platform.  Having PHP access is not critical to your success.  Yes, it can help but you migrated from Volusion which has much more restrictions than BigCommerce (in terms of what you can accomplish from a template perspective and SEO)...so I feel we're missing something here.