Accuracy of basemap on-the-fly projection in NAD27?

[cmako]

I am working on georeferencing some old maps that have coordinates in NAD27 State Plane Arizona Central (FIPS 0202). During georeferencing I set the map projection to this coordinate system so that I can accurately use the coordinate markings on the old maps. 

When georeferencing is complete, there is an obvious offset of the georeferenced map features relative to the ESRI imagery basemap of about 55m NNW. However, if I set the map projection (not reprojecting the georeferenced image) to NAD83 UTM Zone 12, the offset becomes negligible and the georeferencing looks perfect. Maps that are in NAD83 UTM Zone 12 to begin with have the same offset when I set the map/dataframe coordinate system to FIPS 0202. 

My conclusion is that the basemap imagery is actually not being projected accurately on the fly to FIPS 0202. Since there is no offset in UTM 12, I don't think the georeferencing was inaccurate, but more likely that the transformations (NAD27>>NAD83 and WGS84>>NAD83) are more accurate.

Am I correct or is there another explanation?

[RichardDaniels]

On the fly projection is NEVER as accurate as using the Projection methods available within the ArcGIS Pro Data Management toolbox. If this is for a science project, consider using these more accurate methods to transform your data.

Recommendation: find a BM or Survey Station on your NAD 27 map that still exists and then compare the 'projected' coordinates to the published National Geodetic Survey NAD 1983 coordinates for that same point. Note - if you are starting to look at submeter accuracy make sure to consider the transformation year in your error assessment (e.g., NAD 1983 != NAD1983(2011)).

[cmako]

My question is specifically about the Basemap projection.