I agree with slomo and bertsmobile1. You are rolling the dice with that BeeBay carb and adding another variable. What the seller tells you and what is reality are usually 2 different things. Maybe you got lucky for $12.99, but a genuine Walbro C1 carb tested and ready to go is worth more than that given they don't make anymore and there is not drop in sub.
I'd stick to your original plan on trying to get the original carb working with a Zama carb kit. That $12.99 carb could be your backup or just return it for a refund. A lot of sellers are daring you to go through the hassle of returning, just like rebates (most never fill out the form).
Troubleshooting is the the process of eliminating variables one at a time to get to root cause. In reading the thread and advice the carb appears to be the right focus for now IMO. As a DYIer myself I know it can be frustrating, but the pro and advanced DYIer advice you get on this forum is worthy to heed. Statistics doesn't lie. As I said previously I think that tuning a saw and getting it to fire are sometimes the same thing especially for a DYIer stiff such as myself.
You said you had running for 5 minutes so I think that's promising.
Hang in there you are savvy enough to get the puppy going and having fun and satisfaction doing so.