I have a table such as the following:

create table [Thing]
(
    [Id] int constraint [PK_Thing_Id] primary key,
    [Status] nvarchar(20),
    [Timestamp] datetime2,
    [Foo] nvarchar(100)
)

with a non-clustered, non-covering index on the Status and Timestamp fields:

create nonclustered index [IX_Status_Timestamp] on [Thing] ([Status], [Timestamp] desc)

If I query for a 'page' of these rows, using offset/fetch as follows,

select * from [Thing]
where Status = 'Pending'
order by [Timestamp] desc
offset 2000 rows
fetch next 1000 rows only

I understand that the query will need to read a total of 3000 rows to find the 1000 that I'm interested in. I would then expect it to perform key lookups for each of those 1000 rows to fetch the fields not included in the index.

However, the execution plan indicates that it is doing key lookups for all 3000 rows. I don't understand why, when the only criteria (filter by [Status] and order by [Timestamp]) are both in the index.

If I rephrase the query with a cte, as follows, I get more or less what I expected the first query to do:

with ids as
(
    select Id from [Thing]
    where Status = 'Pending'
    order by [Timestamp] desc
    offset 2000 rows
    fetch next 1000 rows only
)

select t.* from [Thing] t
join ids on ids.Id = t.Id
order by [Timestamp] desc

Some statistics from SSMS to compare the 2 queries:

The CTE version seems 'better' at first glance, although I don't know how much weight to place on the fact that it incurs a memory grant for a worktable. (The messages from set statistics io on indicate that there were zero reads of any kind on the worktable)

Am I wrong in saying that the first query should be able to isolate the relevant 1000 rows first (even though that requires reading past 2000 other rows first), and then only do key lookups on those 1000? It seems a bit odd to have to try and 'force' that behaviour with the CTE query.

(As a minor second question: I'm assuming that the last part of the CTE approach needs to do its own order by on the results of the join, even though the CTE itself had an order by, as the ordering might be lost during the join. Is this correct?)