You question is a double edged sword and cuts both ways. There are both Pros and Cons to both the old and new URL structures. Hopefully my explanation helps.
The old structure allows you the chance to keyword focus the page more. For example lets say your keyword is tacos and you are wanting to rule the taco market. The old URL path allows you multiple pages with focused keywords to different long tail keywords. For example you could do tacoking.com/mexico/tacos and tacoking.com/peru/tacos and that type of structure would allow you to pass link juice from the sub page(tacos) to the category page(peru or mexico) and so one. The down side is you could effect the user experience with too many pages and cause a higher bounce rate. Also it sounds like the current platform you use isn't very flexible and might not let you make multiple "folders." If you do use this make sure to plan it out and make it a nice user experience.
On the other hand the new structure makes your website easier to maintain without a bunch of categories and sub-categories causing a vast improvement in the useability of your site. It makes it a lot easier to make multiple pages with custom URLs without researching the category it needs to fit under. It does cause you to keyword stuff your URL alot more than you currently have to.
I hope this gives you some perspective of the different options you have. It sounds like the system you are currently using is really constricted and it isn't allowing you to have the proper freedom to SEO you content how you see fit. There are a ton of open source platforms that you can build a site off of that allows you to do both styles of URLs.Keep in mind that 301 redirects don't pass 100% link juice to the new page it is pointing to and you still have to SEO the new URL you are focusing on. I hope that helps!