Did you try Qwant? https://www.qwant.com/maps

https://foursquare.com/city-guide layers Foursquare's proprietary labeled place taxonomy ( https://developer.foursquare.com/docs/categories ) on top of Mapbox maps which use OSM. It's certainly not all open source data, but if you want something that's an already-deployed website that doesn't use Google tech, it may be your best bet.

https://overpass-turbo.eu/ is a complete search for OpenStreetMap. It’s a bit cumbersome to use, but the wizard is great and the search is super powerful. I particularly like “banks far away from police stations” example: https://wiki.openstreetmap.org/wiki/Overpass_API/Overpass_AP... ;)

GeoNames.org or OSM will give you natural places ("New York", "River Thames", etc). Commercial places ("McDonalds", "Cafes", etc) are primarily proprietary databases.

As a reference, Apple maps sources its commercial places data from a combination of: TomTom, Automotive Navigation Data, Getchee, Hexagon AB, IGN, Increment P, Intermap Technologies, LeadDog, MDA Information Systems, OpenStreetMap, and Waze.

If you can install an android app, I've been using OsmAnd for exactly that.

I do, but I end up using Google Street Maps and satellite view very often. That makes OSMAnd not as useful as it could be. I need at least a satellite view to switch to OSMAnd as my main mapping app.

You can enable the online maps from OSMAnd and enable Microsoft Earth: https://osmand.net/features/online-maps-plugin

It's available for iOS as well.