I just did a bit of research and wonder if there's any advantage to getting the TiVo Stream over the Chromecast or Amazon Fire?
I think you are doing this backwards. You should start by researching the various services to see which one or ones would be a suitable replacement for what you have now. Once you know what services you want/need, it will be easier to figure out which device will let you enjoy TV the way you want to.
I, for instance, mostly watch broadcast television. For me, the TCL Roku televisions are a perfect device. If you plug an antenna into one of these, you can watch broadcast television, use a PSIP sourced (no service or even internet required) program guide, and trick play (pause, rewind, fast forward) if you plug a 16GB usb stick into the television. Plug in ethernet (or go wireless with built in AC networking) for all the apps that come with a Roku -- including Airplay.
Amazon nicely integrates broadcast television with streaming via their FireTV Stick on the set and a Recast whole house DVR. Broadcast and streaming services can be combined in a single program guide. If you just want to build your own OTT service, skip the Recast.
We could walk through a hundred permutations, but it's all mental masturbation until we understand what you want to watch.