So many questions, let's tackle one-by-one.
Recently, Blackberry began selling smartphones that run the Android
OS. If the Android OS has so many security issues out-of-the box, then
why bother paying extra for it?
Because Android has many useful applications which are not otherwise available on Blackberry. Android has approximately 10x more apps, so a chance to find an app you need (for example, Airline-specific app) is much higher on Android. Thus for many users it would be worth paying extra.
Consider that an old cell phone (which is not a smartphone) is extremely secure against many nowadays attacks - no ransomware ever! However you'll see very few people still using it, despite its extremely high level of security.
Blackberry claims that the phones have a "hardened" version of the
Android OS. Are these measures worth the cost, or do they only provide
trivial protection against an experienced hacker?
This is impossible to answer without knowing specifics, especially how contained the Android frameworks are on Blackberry. However it is possible to contain them better on Blackberry, because Android frameworks do not have to run under root or have access to hardware, like they do on Android.
Can you get the same level of protection by simply buying a regular
Android and then hardening it yourself?
Unless by "harderning" you mean "rewrite a significant enough part of Android", the answer is No. You will not have the same level of protection in this case, because you cannot contain all Android code in such a way that nothing Android-specific runs with elevated privileges.