The evidence clearly states that a home birth is just as safe as a hospital birth for a baby and safer for the mother.
Here is the Lancet review on the Home Birth Study:
https://www.thelancet.com/journals/ecli ... 8/fulltext
And here is a great breakdown of the study: https://www.sarawickham.com/research-up ... irth_safe/
We live in a culture that tends to believe that hospital is always the safest place to birth, but that just isn't the case for everyone, so it is absolutely worth you doing some reading and research here.
Here are some great resources:
Why Home Birth Matters - Natalie Meddings
Homebirth: Safe & Sacred - Kim Osterholzer
What's Right For Me? Making Decisions In Pregnancy & Childbirth - Dr Sara Wickham
As regards induction of labour for post term birth and the complications that can arise from this, Dr Wickham has some wonderful evidence based resources on her website: https://www.sarawickham.com/
I would absolutely recommend having a chat to the home birth team at your chosen hospital as they will be able to answer all of the questions that you both may have as well as talk through clearly the many benefits of home birth as well as any of the risks. I really feel that one of the many best benefits of being on the home birth team is the continuity of care that it offers. You will actually know who is looking after you and the women that will be supporting you during your birth.. and that is priceless.
I do hope that this helps a bit.