If all you want to do is document your improvement, then you just need something to measure distance and speed.

HR is a very nice thing to have, IMO, as initially it helped me see when I was expending energy at an unsustainable rate for the remaining distance or hill yet to come on the ride. I've since gotten a better idea of the other physical clues so I pay less attention to HR while actually riding.

As well cadence is nice too. Being a lugger or masher as a kid, I now see and have proof that as I am able to spin faster and maintain a fast RPM, that I can have speed and a slower HR than I ever would trying to mash out speed in the high gears, no matter how strong my legs get.

I don't think it matters much what you get. Phone app or dedicated device. I use a Garmin 500 because I prefer dedicated devices for battery life and the simple fact I can keep my phone safely tucked away. Of course you can still keep your phone safely tucked away and still track your ride and stats, but don't you want to see some of that while riding?

I don't want a map on my device though. I navigate well by intuition and usually just a glance at a map gives me what I need to know for the ride. As well I can pull my phone out and bring up a map the few times it's needed..... weather and radar maps too!

Accuracy... well that's a hard call, all are accurate enough. Comparing any two devices will cause grief though.
The accumulated data from many rides will tell you if you are improving. Looking at any two will not be helpful toward that no matter how accurate the device. One ride is never the same as the next, even when the same route.

Battery life, I think dedicated devices will fare better, though others have disagreed with me. But maybe it's a matter of not leaving the phone screen on so the stats can be seen during the ride. My Garmin edge 500 I can see the screen at all times.

Obsolescence..... it's obsolete the day you buy it. Electronics like this get superseded by new versions with better features very rapidly. However you can still use it obsolete or not. My 500 has been obsolete for quite a while.

But to sum up... it doesn't much matter. Just start recording and saving your data on distance, speed and time. GPS track, cadence, heart rate and other things will give you more wealth of information to consider.

Whether you use an app on a smart phone or a dedicated device it doesn't matter. Just use the same one to track your data with for the sake of the consistency of the data.

For a smart phone, you'll likely need bluetooth sensors. I'd try to buy the sensors that have dual output of bluetooth and ant+ just in case you decide to switch to a device that needs ant+..... or vice versa. I think Samsung phones are some of the few that actually have ant+ so in that case you can just get ant+ sensors, but still it might be wise to get the sensors with both ant+ and bluetooth.