Custom Data Connection Returns JSON But Desktop App Says OData Not Service or Feed

An update here, i.e. answer, for those finding themselves in the same position.  I'll try and be brief so here's the gist:

1.  An out of the box ASP.NET MVC Web API controller returns data when asked for an OData feed, but for whatever reason, Power BI acts like it's no good.

2.  The first part of the fix is to change what your function returns.  In my original connector it was this:

source = OData.Feed("https://www.foo.com/api/data/powerbisummarydata")

Now it looks like this:

DefaultRequestHeaders = [
#"Accept" = "application/json;odata.metadata=minimal", // column name and values only
#"OData-MaxVersion" = "4.0" // we only support v4
];

...

source = Web.Contents("https://www.foo.com/api/data/powerbisummarydata", [ Headers = DefaultRequestHeaders ]),
json = Json.Document(source)

From my reading of things, the DefaultRequestHeaders is not really necessary because Power BI won't recognize it as a feed anyways, but it's here for completeness.

Once this change is implemented, Power BI actually treats it like data, but as a series of "records" without any of the columns associated with them.  So the rest of the fix is done in Power BI itself, after you've deployed your customer connector.

1.  It's best to just start by getting data with a New Query.

2.  The query steps look like this:

//pull the data in from your connector

STEP 1:
= YourSharedFunctionName()  //i.e. MyConnnector.Feed()

//split it into records

STEP 2:
= Table.FromList(Source, Splitter.SplitByNothing(), null, null, ExtraValues.Error)

//expand the fields in your records - greatly simplified here for readability
STEP 3:
= Table.ExpandRecordColumn(#"Converted to Table", "Column1", {"City", "State", "ZIP"}, {"City", "State", "ZIP"})

After Step 3 - voila! - you get rows of data.  Or at least I do; my Web API returns a List<MyDataRecord>.

Hope this helps someone.