Are you particularly anxious about giving birth? Will it make you feel better if the doctor you know is delivering your baby? How do you feel about interventions such as having your waters broken or the syntocinon drip? Do you want to have an epidural? How would you feel about having to have a C-section?
I am also in a country with continuity of care with a particular doctor, and when I had my first baby I was quite anxious and it was really important to me that the doctor I knew and trusted was the one to take care of me during labour.
I ended up being induced at 39+5 for reduced movement, with an unfavourable cervix. Because I had to have the syntocinon drip, I ended up with continuous foetal monitoring and an epidural, rather than being able to stay fully mobile and use the bath tub as I had hoped. I was mostly cared for by midwives I hadn't met before, with the doctor popping in every couple of hours to see how things were going. The induction ultimately failed because my cervix didn't dilate past 6cm and my baby didn't move down, so after a very long day in labour, my doctor called it for a C-section. The C-section itself went very well and I had an uncomplicated physical recovery and I have a small, neat scar. However, the physical recovery in the first couple of weeks when the incision was still fresh was pretty hard, and I found it difficult mentally. This was particularly the case as it is standard practice here for the baby to be taken away during the C-section and not returned to the mother until she is in, or in some cases out of, recovery. I found it hard enough having to wait 45 minutes to hold my son, but most other women who have C-sections here (France) have to wait a couple of hours. I don't know which country you're in but I would ask what the procedure is with C-sections, and in particular whether you will keep the baby with you for immediate skin to skin or not.
In my case, I understand that the induction was necessary, and in fact my son turned out to be very small for gestational age and had clearly stopped growing so it was best for him to be born when he was. I also don't believe it was necessarily the induction which led to the C-section. He was badly positioned which meant that he didn't move down and my labour progressed slowly.
Personally I would not accept an induction before 41 weeks for non medical reasons. However, I know there has been a study in the US indicating that induction at 39 weeks may actually reduce the likelihood of an emergency C-section in first time mothers so it's far from clear cut.
If I were in your position, what it would come down to for me is (1) how important is it for you to have your own doctor there? and (2) are you happy to have a very medicalised birth or were you hoping for something more natural?
There's no right or wrong answer to the first question, but bear in mind that even if your doctor does deliver your baby, they will not be there for most of your labour. The likelihood is that you will spend the majority of your labour being cared for by midwives, with your doctor coming in at intervals to examine you, and then barring any medical emergencies, your doctor will come back when it's time to push (or do a C-section).
Regarding the second question, if you want a natural labour then don't have an induction. It turns labour into a very medicalised process. On the other hand, if you like the idea of turning up on a particular day to have your baby (rather than the unpredictability of waiting for labour to start and having to decide when to go in), and don't mind the idea of having a balloon or a pessary to ripen your cervix, having your waters broken, continuous monitoring, being hooked up to a drip, having an epidural, probably not being very mobile etc, the induction could be a good option for you. Inductions get a lot of hate on Mumsnet, but there are plenty of women who go in to be induced, have a nice epidural, lie on the bed watching Netflix for 6 hours and then push their baby out with minimal pain. If you're one of those women then being induced might be a great experience for you.
The trouble with childbirth is that it's unpredictable. You won't know how it's going to go for you until afterwards.
My second birth was a spontaneous labour and largely physiological VBAC (I had an epidural once in active labour and my daughter was born less than two hours later), which couldn't have been much more different to the first.