It’s honestly not that hard to get the gene set you’re looking for in a single cell. It’s may take tens of thousands of attempts for it to be exactly what they’re looking for but it’s possible with current technology.
The mammoth genome and Asian elephant genome will be much more divergent in non-coding sequence than in protein-coding sequence, so the reference to 99.9% similarity figure in the original post is an inaccurate and highly ascertainment bias. And yes, non-coding sequence differences will matter for phenotype difference.
If they are lucky they will end up with a hairy Asian elephant, but it won’t know what to do or how to survive in Alaska or Finland.
The evolution and divergence of species involves much more than just tweaking a few protein-coding genes. Thinking otherwise without proof of principle in much more tractable species (rodents, flys, arabidopsis, corn) is genetic hubris.