ETA: I thought the repair kit would come with instructions, but it turns out they're online:
iPhone SE battery replacement instructions. Yowza! That is beyond what I want to mess with. I took the phone to a local repair shop to have them do it, and the guy there said if I'm not having trouble with the battery, it's better to wait until Apple issues an ios update that has a setting to turn off their slowdown "feature". I decided to go with his advice, so I'm just stashing my $20 battery & repair kit and calling it a learning experience. Not worth returning the kit since it's $10 shipping.
/ETA
When I updated my first iPhone from ios 7 to ios 8, the voicemail indicator stopped working with data off. I'm usually on wifi, so I have data off most of the time. Since then I have updated the os very reluctantly, since I know it can make things significantly worse. I did update to ios 9, upon which the voicemail indicator worked intermittently (SIGH).
I got an iPhone SE a while back (but I still think of it as my shiny new phone). When I set it up with the backup from the old phone, it was running ios 9, and I left it that way because I hadn't heard anything good about ios 10. When Apple got to ios 11, I figured okay, it's time, and let my phone update. Yeah, the voicemail indicator stopped working entirely again. I just check voicemail every time I miss a call.
I immediately noticed that the phone was running slower, and rolled my eyes about bloatware. However, it turns out that Apple is slowing older phones on purpose, "because the battery is degraded."
Apple Battery Debacle: Yet Another Reason to Support a Right to Repair by Jerri-Lynn Scofield.
Apple is offering $29 replacement batteries, but is having issues with having them in stock. The article recommends iFixit, which is offering
$29 battery replacement kits and does have them in stock. I took a look, and for some reason the iPhone SE battery kit is only $19.99. However, shipping is $11.
I had originally understood that all older phones were slowed down, and it might be useful to get a new battery, but you still have to hope that Apple issues an update that doesn't throttle the phone speed. On second reading to make this post, it sounds like a new battery will solve the whole problem. I sure hope so!
(I'm not affiliated with iFixit in any way, and haven't received my kit yet, but I thought other folks might like to know about the problem and possible solution.)