Warp core stabilisers have drawbacks, almost certainly to prevent their widespread adoption, which would make many engagements much less of a risk for those involved. These drawbacks are decreased targeting range and decreased scan resolution, i.e. increased locking time. Increased locking time will be significant in minor skirmishes, when the engagement can be decided by a few shots either way. A delayed positive lock denies you some early shots and can almost doom you in the engagement before it starts. Decreased targeting range is significant too, denying sniping as a valid tactic, preventing a reply to sniping ships, and forcing a ship to get closer to brawlers.
The drawbacks to fitting warp core stabilisers are significant. They clearly affect how many combat ships are seen fitted with them. However, there are no negative effects for non-combat ships, which is where I believe the real problem with warp core stabilisers arises. The obvious example is with industrial ships, most notably haulers, where targeting range and locking speed are simply not a factor in the ship's role. The only drawback to fitting a warp core stabiliser is that it uses up a fitting slot, but when that slot is not being used for anything else this is also no longer a drawback.
Personally, I see the problem exemplified with the redesigned Tech I haulers that have specialised cargo bays. Whereas previously the low fitting slots forced a compromise on either increasing cargo capacity or adding warp core strength. If you wanted both you would need to train and pay for a deep space transport. But now, if you just want to haul planet goo (or minerals, ore, or ammunition), you can do so with a dedicated bay whose capacity is not modified by modules. Those low fitting slots no longer compromise your fitting, warp core stabilisers have no drawbacks to their use.
There is another class of non-combat ship where the lack of drawbacks for fitting warp core stabilisers is having a negative effect, and these can be seen in faction warfare. The plexing frigates may be combat ships in design, but they have been repurposed to avoid combat entirely. They do not engage other ships, or even intend to engage or be engaged by other ships. For this reason, I think it is no great leap to suggest that these plexers are not combat ships.
The same issues with T1 haulers with specialised bays are true of the non-combat plexers. Any drawbacks designed in to warp core stabilisers have no effect on the ship they are being fitted to, because they are non-combat ships, and so not affected by combat drawbacks, and the fitting slots being used are not diminishing the capability of the ship's primary purpose.
The frustration surrounding the warp core stabiliser is not to do with the module itself, but in its abuse. Clearly the module is not overpowered, as no one ever suggests its use in the same way they do as, for example, the damage control unit. This is because there are significant drawbacks to fitting even one that will have a noticeable affect in just about every combat situation, against rats or other players. The abuse comes when the warp core stabiliser can be used without such a negative effect. Exploitation without penalty.
It seems that to prevent continued frustration with the warp core stabiliser there needs to be a drawback added to the module that will significantly affect its fitting to any non-combat ship. Perhaps decreasing the agility, increasing the inertia of the ship, so that it takes longer to align for warp, giving an aggressor more of an opportunity to rake through the target's defences. But this may not matter given the low fitting slots used for warp core stabilisers are shared with armour defences, and the mid fitting slots for shield defences remain free.
Perhaps fitting warp core stabilisers could lower the resistances of a ship's shields, or reduce the absolute shield HP. Any shield tank would then be compromised, at best only a little above the default levels with the mid-slots dedicated to shield modules, and the armour cannot be augmented without affecting how many warp core stabilisers can be fitted. Or do both, and combine a shield penalty with an agility penalty.
Other, less controversial modules have corresponding penalties to compensate for enhancing certain attribute of a ship. As is the crux of this post, even warp core stabilisers do, effective enough to prevent any serious adoption on combat ships. It is straightforward to consider that stabilising a warp core could effect a ship's manoeuvrability, and manipulating such fields could harm the integrity of the shields. But these are just suggestions, a means to provoke thought and discussion. What should be apparent, though, is that warp core stabilisers on non-combat ships need adjustment.