Let's also not forget about Windows Mobile. The first smartphones powered by it (marketed as "Pocket PCs which also happen to be a phone") started coming out in ~2002, with touchscreen and everything. I own a collection of phones with this OS, including a HTC Wallaby (the phone that the XDA-developers community was originally about), and even that old thing supported true multitasking, mobile data (GPRS), the best Internet Explorer they could put on such things at the time (it was Microsoft, after all), media player, handwriting recognition (inspired by the Palm devices, I believe)... in many ways it allowed one to do "smartphone tasks" well before Android or the iPhone came out. In fact, UX aside, when the iPhone came out, the Windows Mobile users I knew weren't impressed, because in terms of "technological possibilities" there was no news.
I used another more recent HTC device (Blue Angel), second-hand, for a couple of years as my main phone, from 2011 to 2013 IIRC. With WiFi and Bluetooth, decent web browsers available (Opera Mini and Mobile, plus some WebKit forks/wrappers), Google syncing through ActiveSync, for a long time upgrading to a newer phone and OS was not a priority. I only stopped using it for a Android phone two years ago (with the "appification" trend, a web browser is sadly no longer enough).
Coincidentally, the Blue Angel was an early Android target[0], and people have made it run versions up to 2.3 (at which point it's too slow to be usable by today's standards).
I doubt that Android owes much to WM, after all it was a closed-source OS and certainly there wasn't any motivation to do things the "Windows way". It's important to point out that platforms like WM and Symbian existed well before iOS/iPhone OS and Android even had their names decided, and failures apart, the idea of "smart phone" was not the result of solely Apple or Android Inc./Google. Unfortunately, even in many tech circles this is not the idea that prevails.
I also own a Symbian-powered Nokia phone (without touchscreen, forgot the model), and I have no doubt it is a "real" smartphone OS. I can confirm it allows for all that you described and certainly even more in the newer devices with touchscreen.
[0] https://www.youtube.com/watch?v=mu-Yy0phw7U