So let me address some of these points in more specific details. Overall I don't doubt your intentions. It would be good to have a simple to use, component based framework that was hackable and quick and obvious to get up to speed with. But I think the main problem lies with you implementation.

You should not roll your own bcrypt, because it can be easily implemented using core functions, with much less code, that has been peer reviewed, and is secure.

http://php.net/manual/en/function.password-hash.php

Likewise the core DB escaping methods a purpose built, peer reviewed, secure functions. By all means it would be advantageous to wrap these in more user friendly functions, but forgoing them entirely may lead to dangerous situations.

Yes you are right that I could use the templating engine of my choice. But what you are demonstrating is seems insecure by default. I would at least explain how to properly escape output and provide examples to do so. Otherwise it would be advantageous to explain how we would link up twig or blade to a controller, and install them via package manages, and autoload them.

The "really?" was to due to that you would rather skip being secure, then spending what should be an inconsequential amount of CPU cycles, on even a $5pm VPS. If you were at the point of the amount of logins and registrations was putting so much load on your server(s) that the hashing is becoming a concern, it's surely a win and you would hopefully be earning enough money to afford to scale up. Even if not, there would almost always be better places you could optimise first.

You say this was not polished for sharing. Yet you shared it on a major PHP bulletin board. Seems kinda contradictory.

There is no reason why you can't autoload or namespace, even when you are "copying and pasting" libraries. You don't have to use composer to use autoloading. It has been a feature for many years before composer even existed. And as above, by the time that the performance of autoloading becomes a burden on an application, there would have been many more opportunities to optimise.

In terms of coding format, I really don't understand. You use short instead of descriptive variable names. You sometimes put spaces between brackets for if, foreach, return statements, but nor always. You mix spaces and tabs. You mix short and long array syntax. The lack of braces is a choice I understand, but AFAIK it is considered better to always use them.

In terms of namespaces, they are typically used to prevent conflicts with other libraries. When you use the namespace helpers it has a high chance of conflicting. Typically a namespace is composed of a vendor name, and a package name which prevents conflicts.