As an alternate to using a cursor for each DB, could you use:
EXEC sp_Msforeachdb 'INSERT INTO #nonexistent (IndexName,TableName,DB)
SELECT ic.IndexName,ic.TableName,DB_NAME() AS DB
FROM MONITORING.dbo.IDXCHECK ic
LEFT OUTER...

I still think the most likely culprit is that you have 13K files to write to.  How many physical disks are you writing to?  I'd be willing to bet the...

It might not be a bad idea to turn this project of "add a column" into a full re-write of all of the software to utilize a better database design?...

Ah right.  If it runs from the command prompt, it works as you.
Is 'Homebrew01" an AD account or a local account on both machines?
If it is a local...

I'm guessing though that you have some method of connecting to the remote machine via remote desktop or have some way to manage that server apart from just FTP?

FTP will be a similar can of worms with a few fun new problems as you then have to have credentials stored in plain text.  Either that or have no...

I personally disagree with your professor.  While having space for longer things is not a horrible idea, there is no benefit to having it larger than it needs to be....