No, the unifi controller is what it says on the tin. Its where you make all your configuration choices and it 'controls' the devices, it can be run on basically anything including certain Ubiquiti devices (like the express), a PC, a Pi, in a docker container, within a VM or even in the cloud. You don't need to run it 24/7, you can just use the phone app but you loose out on some of the features such as logging and stats.
The express isn't a bad device per say, its just entry level, you've got a router, access point and controller all in one device for £130. In premium network gear world, that isn't a lot of money and you ultimately get what you pay for.
Your plan to use an express and a single access point isn't necessarily a bad idea and will work (you'd still need to power the second AP as the express doesn't have PoE so you'd need an injector or PoE switch), what is not clear is what you are ultimately looking to achieve over and above your current set up.
What's not working for you on your existing setup? (by the way, wanting new nerd toys to play with is more than an acceptable reason
)
I'd definitely suggest doing some more research before spending any money, for example, the access points have specific use cases. The 'disks' are designed to be ceiling mounted, the 'mesh' devices are omni directional and inwall devices are meant to be mounted on/in a plasterboard wall. We would need to know where you plan to place the AP before making a recommendation.
I've got a fairly modern 4 bedroom house which is approximately 132m2, a single access point (U6 Pro in my case) located on the landing ceiling (the middle of the house) provides good coverage in every room (500+mb). I later added 2 more access points to cover the drive and back garden as coverage outside was pretty bad/non-existent. The former is in the detached garage and the latter is mounted on the outside wall of the house, all are hardwired back to the loft where they are powered by a PoE switch.
Fortunately where the internet comes into the house is right next to a gutter downpipe so while the cable up to the loft is ran externally, its not visible. I did have to trench an Ethernet cable out to the garage though.