I disagree, the “share screen” is the logical place for actions that operate on the object you’re currently looking at. Adding a new banner or something for installing PWAs would just clutter up the UI and make it less consistent.
Somewhat off-topic but as someone who used to be a very heavy iPad user (the M1 Mac has kind of obsoleted it), I think of the share sheet as the equivalent of a Unix pipe. It takes your current view or output and makes it available to another process - and if you use Shortcuts, you can actually see the data and its transformations along a literal pipeline.
So, back on-topic, it does kind of make sense in that you’re taking the current URL and sending it to the Springboard.
A perfect menu already exists for this. If you click the “Aa <puzzle piece” button, a bunch of website-specific settings pop up. “Search on Page” or “Install on Homescreen” would fit in perfectly here. Maybe under “Translate Page”.
This is close, but it’s not ideal: the drawer is an OS-wide feature, that puzzle menu is a relic of when the drawer didn’t exist. And its symbol suggests that it’s about text options.