Hello! I wanted to reach out to you because I ran into this same issue. After seeing your review, I thought I should share what worked for me in case it can help you as well.
First, I had to request the memory size on my original hosting platform be increased to its maximum size. After that, I had to deactivate ALL of my plugins before attempting the transfer.
Only then, did the transfer actually work. SiteGround was no help with this.
Hope this helps!
