Why doesn't SQL Server use my index in this SELECT ... WHERE?
Given the size of your table (~11k rows), I think it would be safe to assume that SQL Server estimated that the cost of performing a seek on the non-clustered index and then potentially multiple RID lookups was more expensive than performing a table scan.
There is some evidence to support this theory within the second query plan that you pasted. I would normally expect the Query Optimiser to suggest adding a covering index for your query as you have mentioned in your post. However, it did not. This to me suggests that SQL thinks that doing so would provide little or no improvement over a full table scan.
With all that being said, I am sure that if you added more rows to the table SQL Server may change its mind and ask you to add a covering index or start performing a seek + RID lookup as you expect. If you have Query Store enabled you can always keep an eye out for queries on this table that are causing problems - if it isn't causing you a problem, I wouldn't worry about it right now.