Build a table for tracking time availability with time zones

I need to build data from the ground-up on this project.

I need to track 30-50 people for 6-7 weeks on availability.  I am tracking globally, so for each person I will be setting up a specific table linking "person + time zone" but the table I'm not sure how to build is the days' availability table. 

The things I want to use PBI to do are:

• Confirm I am not scheduling project tasks when an employee is unavailable (this is where time zones matter because if I'm scheduling something 8am-12pm EST, that's 12pm-pm GMT, and 1pm-5pm CET). But I want PBI to tell me if I schedule someone in HKT(Hong Kong) for a 8am EST event (give me an error). Time zones MATTER.
• Shifts are also important because we are dealing with people working adjustable shifts (so Week 1 they may work 1pm-9pm but Week 2 they work 4pm-1am). And see above Time zone concern again.

My instinct (coming in as an Excel-user-learning PBI) is the following. But I see SO MANY PROBLEMS with this. I KNOW this won't work in PBI but I just can't wrap my brain around how to solve for EVERYTHING.

I have no idea how to make entering their time at-all-managable. Either I build columns for every hour of every day (and end up with like 1,200 columns) and make a "yes/no" availability (and I am managing that by hand mind you!!!!) or I can't manage partial days. Example Employee 2 above where they have a meeting in the dead-middle of the day so I can't schedule my project needs during that time!

Basically, I feel like I have no idea how to build this table. This isn't a forever-project, but I really don't want to have to manage thousands of columns/rows/data by hand if I don't have to.  I feel like there must be a smarter way to do this.