The C++ Constructor is very different to the construction script and much lower-level (and they do run during gameplay, they in fact run anytime you instantiate one of those objects in memory).
You do have to be careful what you do in there. Constructors are for initialization only, you shouldn’t be doing any complex logic or trying to reference other classes and objects aside from creating Sub-Objects.
The Blueprint Construction Script equivalent in C++ is OnConstruction(), which you can override on Actors and do more setup if you need to, but really PostInitializeComponents() and BeginPlay() are probably the only things you need. Actor.h has a very long description at the top of the class as to what runs when, and what you should use each function for.
To be honest, Timelines are also kind of a wierd thing to use in C++ - they have an editor tool all of their own for a reason. There’s nothing to say you can’t use them, but it’s like trying to write an entire UMG interface with components in C++ when you have the UMG Editor. As a rule of thumb - if the editor has a bespoke tool for something, you should probably use it that way.