While it's not difficult to modify the library's code to change the default mappings, I wouldn't.
The value of using the full constructor is that it is the only thing that must be changed/modified
for a given LCD interface or wiring.
Once that is done (and it can be only one line) all the rest of the sketch code works the same
regardless of the interface or wiring.
i2c, 4bit, shift register etc....
And that includes backlight control.
For your specific display you could simplify the constructor to use hard coded values:
LiquidCrystal_I2C lcd(0x20,4, 5, 6, 0, 1,2,3, 7, NEGATIVE);
to minimize the lines of code that has to be inserted/replaced.
You have to have a constructor anyway, so would it really save much by modifying
the library to use your particular boards wiring to shorten the constructor?
It would also mean that if you were to get a future updated version of the library, your code
would break until you once again modified the library for your board.
Yes the supplied examples won't work "as is" on your board without modifying the library or the examples,
but, IMHO, the example sketches are not really particularly great examples,
of how to use the library since they don't use the library's newer backlight capabilities/functions.
i.e. once you fully fill in the constructor, there is no need to use the setBacklightPin() or have
to do pinMode() or digitalWrite() functions to control the lcd backlight.
It can be done using backlight() and noBacklight() functions.
For me, modifying the library wouldn't work since I have 3 different i2c backpacks
and they are all wired differently.
On the speed, i2c is not particularly fast. The bit transfer rate is only 100kbit/sec
and there is some amount of i2c overhead per transfer as well.
i2c is great for minimizing pins or sharing pins across multiple devices, but
not so great for speed.
On the brightness, are you doing any clear() operations in a loop?
that can cause flickering or the appearance of dimming.
If not, it sounds like the backlight might be getting proper current.
Although from the photo, the backlight looks plenty bright enough.
Maybe check the backlight resistor.
The i2c module you have doesn't have a current limiting resistor
on it and I can't quite tell if there is one on your board.
Maybe it needs one and the backlight is dim because it is being overdriven.