I have a save right in front of Siddgeir. First quest is to fetch him Black-Briar mead. Second is to kill the bandit leader; however, he keeps trying to send me to Embershard. I've cleared Embershard Mine right at the start of the game. Well over 30 days have passed since I've entered yet it hasn't respawned.

As such, it has rendered the Misc. quest impossible to complete. Now, since I have a save before he gave me the quest, is there a trick where I can get the quest but to a different location?

I would first try leaving and reentering the cell and if that doesn't work I would do some other short quest in a different region and come back.

Also, have you stayed out of the Embershard cell/area for 30 game days? If you reenter the cell the counter restarts.

Good news!

I went into Breezehome and slept for 30 days. While Embershard mine still didn't respawn, the Jarl gave me the quest again but this time in Bilegulch mine instead and I completed it.

Sweet.

I just looked up Embershard (I couldn't remember exactly which one that was.) and it acts weirdly in my game, too. It didn't repopulate but the bridge was raised and the chests refilled.

Same happens to me. Glad I had a save right there.