I was under the impression that because the secondaries are isolated from eachother, it WOULD be ok to do this

In this case, it would be OK. But not necessary. (my opinion)

 

replace the regulator kit with this discrete component regulator.

Why?
I see no advantage of that discrete design, whereas the IC regulator is simpler, more robust, and will give better performance.

 

For the thermal and SOA protections alone, an integrated solution may be the best approach.
Having said that, if you want to proceed with a discrete solution, by all means do. It is a good learning experience.

 

if you want to proceed with a discrete solution, by all means do. It is a good learning experience.

Meaning trying to determine why it doesn't work or why the transistor got zapped.

 

The first place to do any experimenting is ON PAPER!! Draw out the actual circuit, and then exaine it for unintended current paths. To make that method more reliable, avoid using "Ground" symbols, show that actual "ground" connection!
In this thread, already, one of the ideas DID include a difficult to spot shorted circuit, that put one rectifier diode directly across a portion of a transformer. Fortunately it was discovered before being built.