that'd likely be it then.  Running it in compatibility mode you'd be using the old engine for your service broker.
As you had said "According to Microsoft, "nothing" has been...

I know that feeling.  I come here to learn too and help where I can.
I am an accidental DBA as well.

I would give them a high estimate...

If it is currently inaccessable, how much worse can it be?

It will just cause some blocking while it runs.

But 27 GB out of 115 GB isn't...

How large is the table?

Updating statistics on a table will only be slow if that table is huge.  Well, blocking will likely also occur, but if that table...

As a thought - did you have your 2012 DB in compatibility mode when you upgraded to 2016?

Your original post was roughly 2 weeks ago.  

I am assuming that performance is currently slower on that server, correct?

My thoughts at this point would be...

What do you get from:
SELECT count(*) FROM tablename 
WHERE Column >= getdate()-5

I am wondering if instead of doing a DELETE, you could select the rows you want...

If you want to make your licensing costs a little bit lighter, you should look at DxEnterprise.  Same guys who made Polyserve back in the day.  It'll do the failover...