Wizrobe

It's possible to skip any wizrobe mid-fight cutscene by damaging the wizzrobe during its 2 frames of vulnerability right before the cutscene begins, and doing so enough times to kill the wizrobe. However, this can lead to a softlock with the Secret Shrine wizrobe specifically.

Wizrobe as Goron

Punch the wizrobe on the first two cycles, being careful not to get too close too soon and make it disappear. To skip the mid-fight cutscene, time a pound to begin when the wizrobe's head has just hit its leftmost point. Continue holding A after pounding the ground to keep the pound's hitbox active as long as possible. Then back away slightly and begin a full combo right after the wizrobe fully disappears into the portal.

Wizrobe with GFS

Two GFS jumpslashes kill a wizrobe. To skip the mid-fight cutscene, jumpslash when the wizrobe's head has hit its rightmost point and is moving back to the left side.

Secret Shrine Wizrobe "softlock"

first understood by Darkeye and Keeta

It's not safe to go for the normal mid-fight cutscene skip on the Secret Shrine wizrobe because it sometimes will get stuck and never spawn the chest or unlock the door, as seen in the video below. This isn't a true softlock because you can still soar or Song of Time out, but it still costs a lot of time.

This pastebin gives a full explanation of why this happens, but the short version is that before the cutscene, the wizrobe always spawns on an elevated platform, but its internal spawn location is already set to wherever it's going to spawn after the cutscene. When the wizrobe dies, it bounces up into the air but is only treated as dead when it falls and reaches its spawn location's y-coordinate, so if it's killed on an elevated platform while its spawn location is already set to a ground platform, it never dies because the elevated platform stops it from falling enough to reach the ground platform's y-coordinate. The wizrobe's spawn location after the cutscene is randomized among the three spawn locations that it didn't use on the previous cycle, so this "softlock" has a 1/3 chance of occurring if the wizrobe spawned on the ground for the last cycle of phase 1 and a 2/3 chance of occurring if the wizrobe spawned on an elevated platform for the last cycle of phase 1.

Thankfully, no other wizrobe in the game tries to start the mid-fight cutscene from an elevated platform, so this is only an issue for this wizrobe. But this does explain why, after skipping the mid-fight cutscene for the second Snowhead Temple wizrobe or the Inverted Stone Tower Temple wizrobe, the wizrobe can die midair instead of falling all the way back to the ground - this means the next spawn location would have been the elevated one.