I don't know why you say TiVo is limited with its streaming apps? Have you launched the OperaTV store on your TiVo? There is a limited native apps on TiVo but there are hundreds you can add from the OperaTV store of your TiVo.
As far as Roku DVR solutions so you don't have to change inputs:
1. Tablo- Most reliable solution but requires monthly sub fee
2. Simple.TV- You can buy a used gen 1 single tuner unit with lifetime for under $100. However, the company RSS is about to go under and we don't know when they will pull the plugs on their service. (you can read what is going on in their forums: http://community.simple.tv )
3. EyeTV- Requires Roku private channel NowhereDVR and a fast performing Apple PC (Core2Duo or newer model) and ElGato EyeTV tuner. You can use an older Mac fine for DVR recording and watching recording content but Live TV will not transcode good to Roku when running from Mac with older/slower hardware.
4. HD HomeRun with PLEX media server. You can record and watch recordings via PLEX media server but not live TV
5. PLEX- PLEX has a DVR solution but is for PLEX PASS members only. Sorry, don't know much about it.
6. PLAYON/PLAYLATER for Roku rquires PC running PLAYON Media Server.
7. Check out SlingTV, PS Vue and other streaming services that offer some cloud based DVR/catchup options.
But the only 3 solutions that integrate with Roku are Tablo, Simple.TV and EyeTV. With these you don't need to change inputs on your TV. But TiVo is an excellent DVR and streaming device. I have a TiVo Premiere. The only thing I don't like about TiVo streaming is that it is slow performing and loading apps and TiVo is just stand-a-lone for only one TV unless you spend a lot of money on TiVo mini and TiVo stream add-ons which is a deal-breaker for me.
I use Simple.TV and EyeTV for my multi-room via Roku and outside home DVR solution and TiVo on my main TV. Since TiVo supports PLEX and PLEX has a Simple.TV app, I can watch my recordings from my Simple.TV DVR on my TiVo and on my Sony Blu Ray player.
Hope this information and tips help.
I still use my Simple TVs sparingly. I wouldn't invest a penny in it, however. I'm less concerned about recording (I use my Channel Master DVR+ for that, which is connected to our primary TV). I do like being able to watch live TV on a variety of devices. On that score, the Simple TV is already becoming outdated, as it hasn't be updated (or supported in any way) for at least two years. As we now have two Fire Tablets, for which no Simple TV app is available, its usefulness is becoming more and more limited.
Am I correct that you can use a Tablo for watching live TV and for recording (by not scheduling much beyond 24-48 hours) without a subscription? I'm considering one, but also waiting for the new Mohu Airwave for a free live TV solution. We don't watch enough broadcast TV to justify a subscription of any kind.
Some things I want to clarify:
1. You are correct in the fact that Simple.TV is no longer supported and that it is in its last legs but it is still operational and you can get a used gen 1 unit with lifetime (no monthly fee). This is what I use. When Simple goes dark then the obvious substitute is a Tablo but I will hold off on paying sub fees as much as I can.
2. As far as Simple.TV app on your Amazon Fire devices:
You can sideload the Simple.TV app to your fire using the Apps2Fire app
You can search for the apk and install it that way via cloud storage and ESFileExplorer app
You can use the PLEX app and use the Simple.TV app/plugin within PLEX
You can use Simple.TV from your devices web browser and install Wukong pointer app or use a BT wireless mouse for better navigation. (This is how I use it).
You use a DVR+ and I use TiVo but these are standalone options. Only Simple, Tablo and EyeTV offer multi-room and off-site live TV and recording access. HD Homerun you can record to your PLEX Media Server and watch multi-room and off site your recordings but not live TV
For the road, you can watch Live TV using apps like USTVNow, Mobdro, Channel Pear and use PLEX for your recordings. So, with a little imagination and resolve you can still find ways to live without Simple.TV when the day comes but certainly paying for Tablo is the easiest most straight forward replacement.
I like the fact that PLEX has a Simple.TV app so I can watch on my TiVo, Sony BR, Roku's, Android streamers my recordings. So, even if you don't use Simple as your primary DVR you can use it as a recording unit to build your PLEX library. Simple.TV also has an app that allows you to transfer and convert files to other devices. Not sure if Tablo has a similar tool.
Hope this information helps.