Try a new original backlight; the backlights which come combined with keyboards are copy ones and not original. They do not use the same material to evenly distribute the light across the keyboard, so pretty much all of them either just light the middle, or are brighter in certain areas (especially Retina Pro models, where it will be very bright at the top and very dim at the bottom). The LEDs on the older models are in the middle of the backlight, which would explain why all the light is in the middle of your keyboard and not spreading.
Original new backlights will have white dots on the white material (compare to the original one), will have a RU sticker which peels off with the removable layer, and there will be blue tape for the adhesive around the side of the backlight, which will peel off in one piece, not multiple pieces like copies do. New originals are quite difficult to find for the older models, so it is usually best to get used original backlights, so you at least know it won't have this issue.
On the old non-Retina MacBooks, the keyboard backlight LEDs are either on or off; there is no "in between" where only some LEDs will light, unless the fault is with the keyboard backlight itself. The logic board just sends a voltage (KBDLED_ANODE) to 2 different pins of the backlight connector, which do the same thing. You can try plugging the backlight into the logic board loose to see if all the LEDs are lighting. On the newer Retina models, it is possible for some LEDs to not light, since they have 2 different cathode voltages, which cause the top row of LEDs to alternate on and off, but this is not possible on the older models unless it is a keyboard backlight issue, not logic board issue.
In short, you have been sent a non-original backlight where none of them work as well as the originals in my experience. Get an original new/used backlight and try this since the LED light originates from the middle of the keyboard, and the layers in the backlight cause it to be distributed evenly across the keyboard.
EDIT: what an original new backlight will look like https://www.aliexpress.com/item/Best-Qua...
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