Switching regulators like LM2574 , LM2596, LM2576 and so on are ancient designs, outdated, and not worth using unless you already made a product with them and you're too lazy to redesign the board.
You can get much better switching regulators which requires less external components and run at higher frequencies, which means you can use smaller inductors and ceramic capacitors.
The LM2574 requires an electrolytic capacitor on the output because of the properties of electrolytic capacitors, some ESR is required in order to be stable.
For example, AP62305 only requires an inductor and a bunch of ceramic capacitors, and it's factory configured for 5v fixed output voltage:
https://www.digikey.com/en/products/detail/diodes-incorporated/AP63205WU-7/9858424 - or you could get the adjustable version :
https://www.digikey.com/en/products/detail/diodes-incorporated/AP62200WU-7/10491524But yeah, because it runs at high switching frequency you can't use it on a prototyping board, and you need to pay attention to layout (see suggested layout at the bottom of datasheet)
If my application for this is powering a pair of ATtiny[248]5s and... basically nothing else, do I need higher frequency switching, or is 52 kHz plenty?
An ATtiny on its own will probably consume under 5mA ... things you connect to it will often consume more, for example 5-10mA for a LED, stuff like that.
How much inductance do I need? The data sheet recommends 680 µH, but IIUC that gives ~80 mA ripple current. I think my load (recall; two ATtinyX5s) is ~25 mA, which is a little scary, but OTOH I don't really understand what the inductor ripple current means. Another source pointed me at the 10 mH (~5.6 mA ripple) currently in the schematic.
The 80mA ripple current is not the same thing as the output current of the regulator
The datasheet contains explanations :
https://www.ti.com/lit/ds/symlink/lm2574.pdf