Is there a T-SQL command that checks whether the object to which a synonym refers exists and is valid?

That looks like an awfully long script for such a simple check; what's wrong with just:

if exists (select * from sys.synonyms where name = @currentSynonym and object_id(base_object_name) is not null)
begin
  --Add logic here
end

You can use OBJECT_ID with the base_object_name column of sys.synonyms to test whether the base objects exist:

SELECT  [Schema] = sch.name,
        [Name] = syn.name,
        syn.base_object_name,
        [Base object exists?] = CASE WHEN OBJECT_ID(syn.base_object_name) IS NOT NULL THEN 'Yes' ELSE 'No' END
FROM    sys.synonyms syn
        JOIN sys.schemas ON sch ON syn.schema_id = sch.schema_id
ORDER BY [Base object exists?], [Schema], [Name];

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The query was adapted from this answer to this same question.