The reason so many people recommend using ITA Software's "Matrix" system is because it does such a good job of finding the best combinations of fares - if it tells you a price for a combination of segments, then it's almost always the best fare that is available.
Or at least, it does presuming that you buy those segments all on the one ticket! Matrix does not attempt to split flights across multiple tickets, which can sometimes be beneficial - especially when you've got days between flights as you're looking for here. ITA Software was purchased by Google, and forms the basis of the Google Flights search tool which does support flights split across multiple tickets. However in this case even that tool doesn't seem to give better results - but in many cases it will.
Before reading any further, it's worth going and reading the accepted answer to this question, which will explain some of the terms used below (and don't forget to up-vote that answer if you think it's helpful!)
As you've stated, the reason the price is increasing when you add a stopover is because the cheaper fares that are available (R, X or L class, depending on the days I looked) all include the condition :
NO STOPOVERS PERMITTED.
whilst the higher H-class fare includes
2 FREE STOPOVERS PERMITTED ON THE PRICING UNIT - 1 IN EACH
DIRECTION IN PAR/AMS.
Sometimes it may be possible to combine multiple fares to get around such conditions. eg, the fares above are for the trip SFO-JRO. It may be possible to instead price the trip as two separate fares SFO-AMS and AMS-JRO - although obviously that would only be beneficial if the individual prices for those two fares were less than the combined SFO-JRO price. This is referred to as "End-on-end" combining of fares.
Unfortunately in this case, KLMs cheaper fares for SFO-AMS do not allow "End-on-End" combinability, so it's not possible to split these flights into two fares (and even then it appears it would cost more anyway).
This then brings us to the idea of booking separate tickets. Whilst we can't purchase the fares SFO-AMS and AMS-JRO on the same ticket, we can split them into two separate ticket and either book the entire trip as three one-way tickets (SFO-AMS, AMS-JRO, EBB-SFO), or a one-way and an open-jaw return (either SFO-AMS as the one-way and AMS-JRO, EBB-SFO as a double-open-jaw return, OR SFO-AMS, EBB-SFO as an open-jaw return and AMS-JRO as a one-way). This is where Google Flights comes in as it will often find these combinations if they are cheaper - although in this case it doesn't find any cheaper options.
Manually looking at the options above does find a cheaper option - it's it's simply not worth considering. Booking SFO-AMS, EBB-SFO on one ticket, and AMS-JRO on a separate ticket works out around US$9 cheaper than booking it all as a single ticket. This saving is not worth the additional inconvenience that this itinerary could lead to - the most obvious being that if you need to make changes/cancel you'll need to pay for 2 change/cancel fees as you have 2 tickets, rather than just one!
The final option would be to book 2 separate round-trips - SFO-AMS-SFO, and AMS-JRO/EBB-AMS. However because you are not looking to have a stopover on the return, this would mean you would be connecting between two tickets at AMS on the return, and this is dangerous. If your inbound flight was delayed (or had it's flight time changed anywhere in the next 8 months!) you would have no protection from the airline in terms of making your connection. Plus, in this case, it is no cheaper to do it that way - so not even worth considering!