How to save videos on iPhone without installing anything
Safari has downloaded files for years. The reason people think iPhones cannot save videos is that iOS puts them somewhere unexpected.
Safari does download, it just hides the result
Tap a media file in Safari and iOS asks whether to download it. Accept and a small blue arrow appears beside the address bar. That arrow is the entire download manager, and it is easy to miss.
Tap the arrow, then the file name, and iOS opens it in the Files app rather than the camera roll.
Moving a video into Photos
From Files, tap and hold the video, choose Share, then Save Video. It now appears in your camera roll and can be edited or posted like anything you filmed.
This extra hop exists because Photos is a managed library rather than a folder. Nothing is wrong when a download does not appear there on its own.
If the download does nothing at all
Open Settings, Safari, Downloads and pick a destination. When no location is set, some iOS versions quietly discard the file.
Low Data Mode and Low Power Mode can both pause a large transfer in the background. Keep Safari in the foreground until the arrow stops animating.
Why an app is not needed
A downloader app does the same fetch a web page can do, then asks for storage and network permissions in exchange, and often shows adverts on top.
The browser route costs nothing, installs nothing and works the same on an iPad.