I'm a "new" dad from Belgium - we get around 4 weeks of parental leave just after birth and are able to have 4 months full time off, which are free to take until the kid is 12 years old. The first 2-3 weeks are OK, because it's new, need to learn a lot and time flies, but I remember I was pretty relieved to be able to go back to work and do something else than watching the baby after those 3 weeks (even though I don't like my work).
However, I can confirm that staying home and being there with mother and child are precious moments. And if I was in the situation you're in, I wouldn't doubt and also take the 15 months leave. However, what will your situation be with daycare? Will the baby be home all the time or will it go to a daycare at some time during the 15 months? I just ask because being at home with your child 24/24 7/7 is quite intensive and after a while it can indeed give a not very fullfilling feeling (as the "only" thing you do is taking care of the baby). In that case I would suggest to arrange some me-time for each of you. For example 1-2 days per week each of you is "off" and can do whatever you want (work on investing, sports, seeing friends, ...). In short, having time with you child is precious, but you need also to make time for other things after a while.
Related to the 3 months you're alone, I assume this is at the end, so when the baby is +12 months. This is the "nice" period. I liked it the best after 6 months (which I really don't like) and before 18-24 months, as after that they are great but also more difficult (read, more tiring). Again, if you have no daycare during that period, being a full time dad will be nice but intensive. Ideally you at least have 1 day per workweek where you can have me-time (even if it just a day where you don't have any plans, but not having the responsibility to think of the child is needed).
Also a factor to take into consideration (at least in BE) is the season/weather. It's much more easy to take care of your child when it's sunny and warm, because you can find activities to keep them busy much more easy.