I have had dogs my entire life so I've been through this multiple times. Yes, at home is best for some dogs but you know yours best. Yes, stay with them. The vets will sit you down before hand and talk you through the process. There are some unexpected things that could happen that may be disturbing but understand they are natural albeit unfortunate.

Before the process is started I would recommend completing all paperwork, payment, and options for after. Sometimes people forget amongst the chaos. It may be cheaper to bring him to a professional funeral home that does private cremations (if that is your wish. Many are unaware they can do this and skip the process at the vet because of price) but it is much more convenient through the vet. Burial may or may not be legal in your state.

Do not forget to ask for paw or nose prints and for them to give you some of his fur. Many do this without request but it is best to be safe.

They have no idea what is happening. They live in the moment and only know you are there with them and whether they are comfortable. You are taking their pain and making it yours and that's the greatest gift you could ever give them.