homebridge is a translator, nothing more
it translates non-homekit things into homekit's "language", Allowing non-homekit things to "speak HomeKit" and be added to the home app. any scripting or automation is done in the home app, shortcuts, or 3rd party apps like eve.
*there are a few slightly hack-y (I don't necessarily mean that in a bad context) homebridge plugins that will do things like add timers, but it basically adds 2 devices to the home app, a switch that when turned on will start a timer for a pre-set length, and when that timer runs out, it "flashes" a motion sensor. but you then have to configure what to do when that motion sensor triggers in the home app. Or homebridge-alexa which well let you add any devices in homebridge to your Alexa home, devices in your HomeKit home that don't go through homebrige won't be exposed by that plugin. Or another similar that lets you control homebridge devices using a web browser.
homeassitant is home automation software, so it has scripting and such. They have also coded homebridge's function into it. So you can add homeassitant devices to the home app.
Ideally you'd add native HomeKit stuff directly into home assistant, so you could perform more advanced scripting with all of your devices. You would leave all your native devices added directly to HomeKit, and not use HA's translator. Those devices would then take commands from both HA and the home app.
I've also seen node-RED mentioned as an alternative for more advanced scripting, but I'm not that familiar.
I use a program called homeseer, I do have to use homebridge and a plugin to get the devices I want into HomeKit.
I've been using it for about 20 years, Might not be the choice I would make with the other options today, it still works great, just a little pricy.
I've got z-wave light switches, LED tape controllers, motorized shades, door sensors, and temp/humid/motion sensors. All of those go into homeseer and then through homebridge into home
I"ve also got Hue bulbs, which are into HomeKit natively, and into home seer through a plugin. Any change from one form of control shows up in the other. (I ask Siri to turn the light on, I can see it turn on in homeseer)
I've also got a couple dummy devices in homeseer, like a switch that says if I'm home or not, and a home "alarm". the alarm doesn't alert anything, it's more of a switch to set the state of my house.
how things are controlled/automated depends on what makes sense, or the capabilities of the controller
scenes are pretty much all done in home
--the home/away switch is triggered by basic location based automations in the home app
-- I have a light behind the TV, and use home app to turn that on, since I can't see the state of my TV in Homeseer
the rest of my automations are done in homeseer, since it's interface is much easier to deal with, and I can get more complex if I need to.
--I have an automation setup in homeseer that will turn on a couple lights for the cat if I'm away and it's after sunset. (not really possible in HomeKit, since HomeKit needs a trigger, and not just the state of devices)
so basically I use homekit/Siri for basic control of lights and scenes, any automations or things that require a bit of logic happen in homeseer. having all devices available in both makes that possible.