Nope. IOS manages its own ram and shuts down apps long before lack of memory comes anywhere close to affecting the system. Your iPhone will NEVER "shut off" due to having too many apps open, because IOS will never leave more than a few apps open under any circumstances. This isn't my opinion, it's fact.
The majority of apps you think you're "closing" in the app manager are actually dead already, frozen in a completely passive state. All you're doing is removing them from a list of recent apps, and freeing up virtually zero ram.
If your phone is "slowing to a crawl" there's either something wrong with it or, more likely, you have one or two rubbish apps that are causing you specific problems. Do some detective work and get rid.
To add to what you're saying on the flip side when Android starts to hit under 100MB of free RAM the entire system starts to bog down. This is why people habitually do this in Android and when they come over to the iPhone they continue that ritual. I've see my iphone hit 50MB free RAM and the system doesn't even stutter.