Does sour milk have something else in it besides lactic acid that gives it such a horrific taste and odor?
Probably. Lactic acid bacteria produce other volatile compounds such as acetaldehyde and diacetyl that could be perceived as off-flavors. Both where fermentation is the objective and in the case of spoilage, the VOC profile can vary with the identity of the responsible bacteria but also of the milk and its treatment.
For instance, in Ref. 1 they discuss the effect of different lactic acid bacteria:

During fermentation, S. thermophilus and L. delbrueckii ssp. bulgaricus generate lactic acid and a variety of volatile organic aroma compounds. These compounds impart particular flavors to fermented milk (Beshkova et al., 1998). The overall flavor of any fermented milk product is formed by a large number of these volatile compounds. Among these compounds, carbonyl compounds and organic acids such as acetaldehyde, diacetyl, and acetic acid play a crucial role and can be used to evaluate the flavor quality of dairy products (Zha et al., 2015). Ott et al. (1997) found that acetaldehyde provided the typical flavor of dairy products such as yogurt and buttermilk. Erkus et al. (2013) also reported that diacetyl contributed the characteristic sour buttery flavor to some fermented dairy products.

The offending chemical could also be vinegar (acetic acid). Again from Ref. 1:

Acetic acid has been found previously as a volatile in yogurt by Panagiotidis et al. (2001). High levels of acetic acid contributes most to the tart flavor of yogurt and is particularly associated with a “vinegary, pungent, acidic” taste (Panagiotidis et al., 2001).

If you search for &quot;chemical profile of spoiled milk gc&quot; you'll find a wealth of references discussing the effect of different milk (due to species, handling or forage), milk treatment, and spoilage bacteria on the VOC profile.
In addition, what makes the milk taste bad may depend on the bacteria and on you. That makes narrowing the answer down to a particular compound impossible.
References

T. Dan, D. Wang, R.L. Jin, H.P. Zhang, T.T. Zhou, T.S. Sun, Characterization of volatile compounds in fermented milk using solid-phase microextraction methods coupled with gas chromatography-mass spectrometry, Journal of Dairy Science, Volume 100, Issue 4, 2017, Pages 2488-2500, ISSN 0022-0302, https://doi.org/10.3168/jds.2016-11528.