The Ubiquiti access points have their own hand-off process, and don't require a controller for them to function adequately. I think that Aruba Instant On access points function similarly.
There are some advanced features that would require either a hardware or software controller, but for basic use - you don't need either.
(Though you do have to install and run their software controller for the initial setup of the access points. After that, you can close it down, and only need to run it to check for firmware updates, or make changes to the access points.)
I believe the dream routers come with their network suite built in, so you're probably good there. I will point out that, if you're using the "Dream Router" (UDR) -- the cylindrical router -- it does cap out at 700Mb routed speeds. . . so if you're planning on 1Gb speeds from your ISP. . . the UDR won't handle those speeds.
There are other routers in the dream series, such as the UDM Pro, UDM Pro SE, etc. . . but those would be a overkill on a 1Gb service, and they don't have built-in WiFi, so you'd end up needing 2 access points, instead of one.
Most of the more affordable prosumer routers are wired-only. The EdgeRouter-X, TP-Link ER605, and the TrendNet TWG-431BR are affordable, but will cap out around 925-940Mb, give-or-take.
What's going to help with multiple smart home devices isn't so much the router, but the WiFi source that they are connected to.
The so-called high-density access points will probably have the biggest impact on your performance.
The TP-Link EAP620HD (1Gb LAN port) or TP-Link EAP660HD (2.5Gb LAN port) can supposedly handle a ton of clients.
Unifi also has some access points branded as "HD".