Change Google maps to OpenStreetMap

Something like yes.

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Change map using this icon:

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But OSM on Connect is a copy served from https://connectosm16.azureedge.net/... so changes made to OSM will be visible here much later. And you can't choose map style.

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A way, to see other map style and up-to-date OSM - is to export an activity to GPX file and open in additional program like:

JOSM 15238:
https://josm.openstreetmap.de/

Using JOSM, you can also edit OSM and make maps more accurate - adding missing paths. Which can be done easier, as you can see your GPS track on the map.