Instead of going around Eldin and taking the long way to Lanayru it is possible to clip through the gate on Hyrule field by using Epona.
By bonking the gate with either a B-Attack or a Jump Attack and then sidehopping into the gate as wolf it is easy to clip through after a few tries. A B-Attack is recommended over a jump attack, to execute this trick have a slight angle away from the gate as shown in the video so when you hold left while targetting you get the action command to "Jump" which is sidehop. Once you have the angle, hold target and left then press B to B-Attack the gate, after this start mashing A while holding left to sidehop, which a small amount of luck you should clip through. In a run you would either ignore the bokoblins by dashing past them and quickly targeting or kill them with B-Attacks and a spin.
The original method was to clip through the gate by bonking into the gate and then dashing through on the right frame. This is insanely hard to pull off RTA as it is frame perfect, the frame can vary every time and sometimes might not be possible due to the RNG of the gate swinging.
If you setup Epona's alignment properly you should either:
After voyaging part-way into the province, you will be warped to the Great Bridge of Hylia, by the Twilit Bulblin cutscene.