pH remains unchanged when the buffer is diluted slightly
This is a good approximation for relatively diluted solutions with low ionic strength only, but when you change ionic strength from 1.0 to 0.1 this rule may give even an unpredictable result.
the pH of the buffer reaches 7
This is not always true. A high dilution of some buffer solutions may result in exceeding the pH of 7.
Working with ionic strength ~0.1 M, a row error of 0.05 pH unit is obtained. Such an error is acceptable at the beginning of studying chemistry.
But using ionic strength 1.0 you are going up to advanced chemistry level.