From personal experience custom engines offer more flexibility, assuming they're not 20 year old monolithic behemoths that tens (or hundreds!) of developers added their little piece and nobody bothered to remove much over the years. IF they're not that, it becomes easier to mold them to what your game(s) need (especially if you focus on similar genres, at least from a gameplay/engine requirements perspective).
Also from a tools perspective, big generic engines like Unreal and Unity have their established ways of working internally that may make writing good tools either hard or suboptimal (meaning that you can do something but since it isn't how the engine likes things it would not be fast). It is usually good enough for most uses, but if you want to make something very specialized you'll have a better time doing it with an engine you have control over, otherwise it'll be an uphill battle (there are cases with Unity for example of people having tools outside the engine and just import the final data, using Unity as little more than a scriptable renderer with asset manager and physics).
On the other hand, custom engines mean custom tools and more often than not tools get the short end of the stick in custom engines. It is a double edged sword, on one side you can make some simple and great tools that focus exactly on what your game is all about (something like Unreal is similar to C++: there is a lot of stuff that you can ignore but they still provide mental overhead even when you ignore them), but on the other side you have to make tools (and just in case, using Blender or 3ds max or whatever as your level editor is the perfect road to inflict pain on everyone involved, often leads to low performance with little visibility on why - and unless you have a vendetta against level designers do not even consider using dummy objects for level scripting, the only thing worse than that is asking people to paint triggers on a grayscale terrain texture using Photoshop).
(note that i'm assuming you have access to source code here which rules out Unity for the majority of developers using it - and of course i also assume that you have the know-how for working on custom engines)