How can I improve my Georeferencing results?

Question

Background This is my second question related to georeferencing naked raster maps in order to re-visualize them on different coordinate systems and in conjunction with other data layers. The previous question is at Convert an arbitrary meta-data-free map image into QGIS project

Problem My goal is to georeference this map:

This does not appear to be Plate-Carrée. So in QGIS, I created several reasonable control points, which for completeness I have attached at the bottom [ref:1]. I provide QGIS Georeferencer the same target SRS as my project file, EPSG:4326. I get exceptionally poor results with Helmert and the polynomial transforms but get a reasonable image with thin plate spline (which makes the resulting geoestimate go through my control points). However, even this result is poor, e.g., at higher latitudes (see the Russian coast north of Japan). This is a screenshot of my QGIS screen using a Natural Earth background.

Alternative path I tried a similar exercise with the much easier-to-use tool at MapWarper: see the result and control points at http://mapwarper.net/maps/758#Preview_Map_tab where I get poorer results (probably due to the fact that I added fewer control points).

Questions in nutshell

1. Are there are any tricks I'm missing to getting a good georeference?
2. Is this projection instantly recognizable?
3. At Unknown Coordinate System on old drawing, gdaltransform is suggested to transform several coordinate points into a several target SRS, with the goal of actually uncovering the projection parameters used to generate the original map. I tried something like this: after saving my QGIS list of points, I did some string processing to get a list of space-separated long/lats via cat eurasian-steppe-gcp.points | tail -n+2 | cut -d, -f1-2 | sed 's/,/ /'> tmp.txt and inputting the resulting file into gdaltransform: gdaltransform -s_srs EPSG:3785 -t_srs EPSG:4326 < tmp.txt and switching the s_srs and t_srs flags (the project uses EPSG:4326). I know I'm shooting in the dark, hoping to get lucky, so I wasn't surprised when I couldn't make sense of the outputs. Can someone expand on how I would use this method to find the best estimate of the source map's projection and projection parameters? My thinking behind this is that rather than messing with placing myriad control points for a good georeference, might it be easier to get a near-perfect georeference with fewer control points, just looping through all the common coordinate systems? Does it involve cross-validation of each point against all the others, for each CRS under test?

I'd like to get an understanding of either this algorithm or of georeferencing so I can automate the process---I run into this issue all the time, and until content creators stop treating their maps as one-off creations never to be integrated with other content, I don't expect to stop.

References

[ref:1] QGIS GCP file:

mapX,mapY,pixelX,pixelY,enable
142.632649100000009,54.453595900000003,505.941176470588232,-95.220588235293974,1
154.934252200000003,59.559921699999997,536.411764705882206,-52.779411764705742,1
80.080158100000006,9.657192300000000,291.558823529411711,-322.661764705882206,1
10.448442600000000,57.819128900000003,21.676470588235190,-103.926470588235134,1
34.007173000000002,27.761438299999998,101.117647058823422,-244.852941176470466,1
50.950890399999999,11.862196600000001,171.852941176470495,-313.955882352941046,1
29.713217199999999,60.024133200000001,90.779411764705799,-92.499999999999829,1
60.000000000000000,0.000000000000000,208.308823529411683,-362.382352941176350,1
69.867506500000005,66.639146199999999,224.088235294117567,-33.191176470588061,1
27.276107100000001,71.049154799999997,89.147058823529306,-21.764705882352814,1
140.000000000000000,0.000000000000000,536.955882352941217,-362.926470588235190,1
20.000000000000000,0.000000000000000,43.441176470588132,-362.926470588235190,1
20.196882700000000,31.243024100000000,47.249999999999901,-231.794117647058698,1
9.171861099999999,42.848309999999998,8.073529411764603,-175.205882352941046,1
131.955786100000012,43.196468600000003,481.999999999999943,-162.691176470588090,1
73.813303700000006,45.169367200000003,256.735294117646959,-161.602941176470438,1
50.602731800000001,44.589102900000000,168.044117647058727,-167.588235294117510,1
121.394975900000006,18.941421099999999,455.882352941176407,-284.029411764705742,1
103.987047000000004,1.417439300000000,389.499999999999943,-357.485294117646959,1
109.325478599999997,55.962283100000001,380.249999999999943,-98.485294117646902,1
31.454010100000001,46.562001500000001,95.132352941176379,-158.882352941176322,1
43.639560299999999,68.844150499999998,137.573529411764611,-40.264705882352814,1