There are two primary movements involved in this exercise: hip flexion and shoulder hyperextension.
Hip flexion is performed by the gluteals and hamstrings, with the spine being supported isometrically by the erector spinae. Relative weakness of the gluteals is common, but highly unlikely to inhibit the performance of this exercise entirely.
Shoulder hyperextension is initiated by the latissimus dorsi, teres major, triceps brachii (long head), and posterior deltoid, with the medial and lateral heads of the triceps assisting the long head in stabilising the arms in a straightened position. The trapezius further enables the posture by ensuring that the shoulder girdle is and remains retracted. However, in the latter part of the movement, all of the shoulder extensors—the latissimus dorsi, teres major, posterior deltoid, and the long head of triceps—approach the end of the range in which they can effectively develop tension. And this means that further shoulder hyperextension can only occur passively, driven entirely by hip extension, and supported by the two single-joint triceps.
Thus, the limiting factor here, as you surmised, is most likely the triceps brachii—specifically, the strength of the medial and lateral heads. They are required not only to extend the elbows under your weight, but to do so resisting the horizontal component of your weight force which cannot be resisted by the shoulder extensors.
The solution, therefore, is to strengthen the medial and lateral heads of the triceps. This should be done performing a combination of both compound (e.g. close-grip or ace-of-spade push-ups) and isolation (e.g. push-down) exercises. And of course it would only help to supplement those with exercises for the posterior chain, which might very well include ‘Short Bridges’.
In terms of body-weight exercises, bench dips would be a good choice since they approximate the joint angle of the shoulders (i.e. hyperextended). And there are a number of variations—bent-, straight-, and raised-leg—which would offer a nice progression. However, the load of bench dips is ultimately limited by the fact that the legs are supported. It is worth remembering, therefore, that it is not necessary to isolate the medial and lateral heads, but only to train them. A regular dip from parallel or diverging bars solves this problem by giving us the full weight of our body to work with.
I hope that is helpful.