It's an interesting question. This 2010 paper contains some discussion of the function of the hyoepiglottic ligament. The following is a quote from the introduction:
The epiglottis is connected to the
hyoid bone by the hyoepiglottic ligament (ligamentum
hyoepiglotticum), which is a connective tissue structure that
serves as the roof of the paraglottic space and the
preepiglottic space. Anatomically, the hyoepiglottic ligament
separates the supraglottic larynx from the base of the
tongue. Some clinical studies have shown that this
ligament inhibits the spread and invasion of carcinoma... The hyoepiglottic ligament is elongated by
approximation of the thyroid to the hyoid during swallowing.
Tension on the ligament results in a “snap” retroversion
of the epiglottis. Thus, the folded epiglottis represents a
state of higher energy than that represented by the unfolded
epiglottis; relaxation of the tension on the hyoepiglottic
ligament releases the cartilage to automatically return to its
normal resting position [emphasis mine].
In the study, the authors demonstrate histologically that the hyoepiglottic ligament is bipartite, with a part they call the pars hyoideus connecting the anterior epiglottis to the hyoid, and a part they call the pars lingualis connecting the epiglottis to the root of the tongue (they claim that this attachment to the tongue was previously unreported). Another quote from the paper follows:
It is suggested that the pars lingualis plays an
important role in active mechanisms such as upward
movement of the epiglottis during phonation associated
with tension of the lingual muscle and supraglottic airway
patency during respiration.
To get some idea of how tension in a ligament anterior to the epiglottis can cause posterior movement (i.e. closure of the epiglottic 'flap'), I found this 1979 paper useful. They used cinefluorography to investigate the biomechanics of the epiglottis. The following lengthy quote should help to clarify the proposed model:
The radiographic observations indicate that
at the peak of the deglutitional excursion of the
hyoid bone and the larynx the following circumstances
(1) Forward translation of the hyoid bone in
conjunction with extreme thyroid-hyoid approximation
causes extreme elongation and
tension on the hyo-epiglottic ligament at its attachment
to the margins of the epiglottic
(2) The compression of the median thyrohyoid
fold, secondary to extreme thyroidhyoid
approximation, tends to make the epiglottic
tubercle bulge backward transversely
as well as vertically.
The principle of the mechanism of folding
deduced from these observations is demonstrated
in a working model that can be set up
in a few seconds. All that is needed is a rectangular
piece of fairly stiff paper. The piece
of paper represents the complete epiglottic
cartilage (epiglottis and tubercle). A $\Lambda$-shaped
crease is applied at the center (Fig. 4) to simulate
the groove on the back of the cartilage,
between the epiglottic tubercle below and the
epiglottis proper above. One next applies stress to the model from the opposite side, at
the site of attachment of the hyo-epiglottic
ligament. This is done by grasping the paper
lightly at the edges, below the level of the
crease, whereupon the part above the crease,
representing the epiglottis, immediately begins
to fold down.
Each limb of the crease functions as the
directrix of a cone; the apex of the crease is
the vertex of the cone. When the paper is
gripped the directrices rotate towards each
other and pull the attached paper after them,
giving it a conical shape.
I've attached the figure 4 from the paper to illustrate better (I'm not sure if that's allowed on SE; please delete it if that's appropriate):
$\hskip 2 in$
The authors of this paper and this paper argue that the model just given does not fully explain the motion of the epiglottis during swallowing. The latter paper describes the existence of distinct lateral hyoepiglottic ligaments attaching to the ends of the greater horns of the hyoid bone. These, they argue, transmit the biomechanical forces generated by the movements of the hyoid and thyroid cartilage to the epiglottis to cause downfolding, although their full argument is too long to reproduce here.
Hopefully all of that has shed a little light on the possible function of this ligament. It seems to be quite complex, but there appears to be a relative consensus that the transmission of tension via this ligament is important in causing the folding down of the epiglottis during swallowing.