Embodied cognition basically has not a single paper able to be replicated [0], so it's out the window.
This isn't to say that ToM is a flawed theory - in general, it's conclusion is probably true. Rather, psychology has some terrible, terrible, mainstream practices - things like outlier elimination and p-value rounding basically make it impossible to replicate a study unless the original author is involved.
But _assuming_, without evidence to the contrary, that the theory is still sound is denying the problem exists. Thankfully there is some research [6] attempting to show where the world is at with regards to this particular theory.
The study the article is about uses the fairly typical false-belief task. We've been using versions of it since '83, so it should be solid. Except it might not work at all when a person has autism [1], and as autistic people generally are aware of others and that others can differ in thought, the task may be flawed or our understanding flawed. Or maybe autism really does mean you lack something that fundamental [+]. There are other inconsistencies with it. [7]
The false-photograph task was developed in part because of the apparent limitations of the false-belief task. Unfortunately, Woodward's results haven't yet been replicated, just relied upon.
Before the crisis hit home, the research was leaning towards people with autism being ToM deficient, but replications don't show statically sound results that say it conclusively or not, thanks to p-value rounding, whilst some more recent research suggests we simply aren't measuring it correctly, and those results may be caused by the coping mechanisms employed. [2][5] Which, if true, suggests that the false-belief task may not actually be stimulating the right parts of any individual's mind, but rather just engaging them in something visually intensive.
A lot of our measurements of ToM, such as when using fMRIs, have been called into question, as they might just be falsely noticing the spatial orientation that happens during visual stimulus. [3] But again, there isn't enough replication to say definitively one way or t'other.
There's also some lesser issues. Some of the strongest ToM research suggests that the origin lies in mirror neurons [4], but the animals studied in the article of this thread don't have them, and animals possibly having ToM is extremely inflammatory within the broader psychological community. If you want someone to try and replicate your study, suggest that a particular animal has ToM. There's about a 50/50 success rate, which is not really encouraging.
Hopefully that's a little bit to chew on. As I've said, ToM probably does work. But we don't currently have the statistics to say it does.
[0] https://qz.com/1525854/psychologys-replication-crisis-is-deb...
[1] https://www.sciencedirect.com/science/article/pii/0010027785...
[2] https://www.nature.com/articles/nn1611
[3] https://doi.org/10.1177%2F1073858407304654
[4] https://doi.org/10.1016%2FS1364-6613%2898%2901262-5
[5] https://doi.org/10.1016%2Fj.neuroimage.2011.02.067
[6] https://www.sciencedirect.com/science/article/pii/S235234091...
[7] https://journals.sagepub.com/doi/full/10.1177/09567976124478...
[+] I'm not the right person to judge this. I have autism. Which, if the ToM theories are correct, means I lack the perspective to tell whether or not I have a ToM deficit.