so the only suitable reason I could think of is that along an electric field line, no matter where ON IT, the electric field strength due to that ONE line, will remain constant.

You can't think about the electric field strength as being "due to a line". You have to think of the electric field strength in terms of the density of the lines.

The density of the electric field lines between the two parallel plates is constant, and thus the field strength is constant, because the field of a single charge on a plate is added to fields produced by all the other surrounding charges on the plate to make the overall field, and density of field lines, uniform between the plates.

Hope this helps.

so the only suitable reason I could think of is that along an electric field line, no matter where ON IT, the electric field strength due to that ONE line, will remain constant.

You can't think about the electric field strength as being "due to a line". You have to think of the electric field strength in terms of the density of the lines.

The density of the electric field lines between the two parallel plates is constant, and thus the field strength is constant, because the field of a single charge on a plate is added to fields produced by all the other surrounding charges on the plate to make the overall field, and density of field lines, uniform between the plates.

See the answers to the question posed in the following link:

Proving electric field constant between two charged infinite parallel plates

Hope this helps.