Use of integer instead of interval (of one type) Our DB design presently has a interval column which will only be storing days (no other interval type) so it is making sense to use INT2 (smallint) instead of interval. Reference to documentation.

Advantage: 2 bytes instead of 12 bytes (we have many such columns).

Is this line of thinking ok or am I overlooking something?
Perfectly ok - especially since you can add and subtract integer to / from  date directly in Postgres.

However, int2 might not be better than a plain integer. True, integer occupies 4 bytes instead of 2, but many operations are optimized for integer. Among other things, integer is the default numeric type for numbers without decimal point. For int2 you often need to add an explicit cast to avoid errors.

Also, while we are talking about a single column, you will most probably gain nothing in regard to disk storage or RAM. Some closer understanding of storage mechanisms is needed here, in particular padding and data alignment.

If you have several int2 columns in a table  (or even "many" as you mention) and / or know what you are doing, the odds for int2 get better.

Details:
Configuring PostgreSQL for read performance
Calculating and saving space in PostgreSQL

If in doubt, run a test and measure:
Measure the size of a PostgreSQL table row