Some sources online (e.g. this, see section "Effect of Earth’s Eccentricity") claim that if the earth's axis was not tilted, then the analemma would have the shape of an ellipse/oval. I think if there was no tilt then the midday analemma would simply be a line along the celestial equator from any latitude (i.e. sun would have has constant elevation at midday throughout the year). And if the analemma was not taken at midday, then its shape would be close to a line with constant celestial latitude: there would be some small variation in solar elevation due to the true solar time differing from the mean solar time (the analemma is taken at the same mean solar time every day). Am I missing something or are those sources simply inaccurate?
The timeanddate.com article says:
If the Earth’s orbital path was elliptical, but its axis not tilted, the Solar Analemma curve would be oval shaped. At the Equator, this line would be a straight line spanning from left to right or West to East.
In a zero-obliquity scenario, an observer at the south pole would see the Sun's declination vary from +8.6" at aphelion to +8.9" at perihelion. The analemma's east-west extent due to orbital eccentricity would be about 3.75°, a factor of 45000 larger than its north-south extent of 0.3 arcsec. To me that's more like a line than an ellipse.
The effect of ecliptic-aligned Earth rotation on solar declination would be negligible. The analemma would run parallel to the celestial equator regardless of the time of day it was photographed.