Did the inspector give you an idea of life left on the roof? Were there any spongy or soft spots? Do you know if there is a different roof type under this one and lastly what year was the house built?
Regardless of that, living in Oregon myself I know we get 265 days of precipitation every year so our roofs do not last as long as some places and gutters are mandatory. It looks like your place had some sort of integrated drain system built into the roofing system, where water was routed to those (small) drain holes. One thing I do not like about that system is that it will have standing water in the integrated channels for the drains. Also if it rains a lot, or if you have any debris in your downspouts you will have overflow problems.
The other thing I do not like about that system is that the rain is run through the drains which are holes in your roof, through the overhang and hopefully down the downspout. The reason I do not like that is because you could have a small leak near the hole and the water will run inside the overhang rotting it out, even worse is that it could run into your house causing dry rot there.
So, I would suggest having a roofing contractor come out to determine useful life on the roof (unless you are comfortable with the one your inspector gave you). then if he says there is at least another 5+ years I would have the drain holes patched over, the low spots filled back in so all of the water will flow off the roof, and then have a new gutter installed, the standard type we have around here. I would have them replace the downspouts, it may be the photo but they appear to be small and can become more easily clogged. It looks like there is a stand of trees around your place so you are going to have plenty of gunk built up in the gutters and or downspouts.
Rubber roofs can last a long time if maintained properly and that is what is used most of the time on commercial buildings with flat roofs here in Oregon so if yours is in good shape you may have a good system in place. If the slope is too low you are better with a roof like yours than a composition shingle roof because they require enough slope to make the rain go down and not into your house. A metal roof may give you better performance and life span but they are more expensive, so if yours works why replace it.
I hope that helps,
Ex Realtor, house flipper, remodelor and now building a vintage truck, making friends and fixing my house.
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