Jack Hargreaves (1960s country/diy guy) method of disassemble hide glue was to drill small hole and insert plastic tube and connect to boiling kettle steam, soften the glue ease joint apart & easily scrape away old glue
At 1:50 you ask what the best repair is. My suggestion focuses on the dog.
Strewth,so much damage, new legs & rails or do you rhink 6oy could splice/laminate and blend in ?
There is, I was quite shocked. I'm erring on minimal intervention with the legs, and a solid repair rather than invisible. They will tell a story.
How long was the dog in the animal hospital after destroying his master's lovely furniture?
The dog likes the taste of hide glue.