Why can't I simply add an index that includes all columns?
First of all, an index in SQL Server can only have at most 900 bytes in its index entry. That alone makes it impossible to have an index with all columns.
Most of all: such an index makes no sense at all. What are you trying to achieve??
Consider this: if you have an index on (LastName, FirstName, Street, City), that index will not be able to be used to speed up queries on
FirstNamealoneCityStreet
That index would be useful for searches on
(LastName), or(LastName, FirstName), or(LastName, FirstName, Street), or(LastName, FirstName, Street, City)
but really nothing else - certainly not if you search for just Street or just City!
The order of the columns in your index makes quite a difference, and the query optimizer can't just use any column somewhere in the middle of an index for lookups.
Consider your phone book: it's order probably by LastName, FirstName, maybe Street. So does that indexing help you find all "Joe's" in your city? All people living on "Main Street" ?? No - you can lookup by LastName first - then you get more specific inside that set of data. Just having an index over everything doesn't help speed up searching for all columns at all.
If you want to be able to search by Street - you need to add a separate index on (Street) (and possibly another column or two that make sense).
If you want to be able to search by Occupation or whatever else - you need another specific index for that.
Just because your column exists in an index doesn't mean that'll speed up all searches for that column!
The main rule is: use as few indices as possible - too many indices can be even worse for a system than having no indices at all.... build your system, monitor its performance, and find those queries that cost the most - then optimize these, e.g. by adding indices.
Don't just blindly index every column just because you can - this is a guarantee for lousy system performance - any index also requires maintenance and upkeep, so the more indices you have, the more your INSERT, UPDATE and DELETE operations will suffer (get slower) since all those indices need to be updated.
You are having a fundamental misunderstanding how indexes work.
Read this explanation "how multi-column indexes work".
The next question you might have is why not creating one index per column--but that's also a dead-end if you try to reach top select performance.
You might feel that it is a tedious task, but I would say it's a required task to index carefully. Sloppy indexing strikes back, as in this example.
Note: I am strongly convinced that proper indexing pays off and I know that many people are having the very same questions you have. That's why I'm writing a the a free book about it. The links above refer the pages that might help you to answer your question. However, you might also want to read it from the beginning.
I'm considering instead simply adding an index that includes ALL columns of the table.
This is always a bad idea. Indexes in database is not some sort of pixie dust that works magically. You have to analyze your queries and according to what and how is being queried - append indexes.
It is not as simple as "add everything to index and have a nap"