>from one direction, stopping at a median, and then crossing further over.

This assumes a median, which is not present at most smaller roundabouts in the US.

In Waterloo Region I used to cycle through multiple intersections that were "upgraded" some years ago from conventional stoplights to roundabouts and imo it was a huge downgrade to my sense of safety. I went from having a clear right of way (hand signal, cross in the crosswalk) to feeling completely invisible to cars, essentially dashing across the road in the gaps in traffic as if I was jaywalking.

I could handle it as an adult just walking my bike but it would be a nightmare for someone pushing a stroller or dependent on a mobility device.

Roundabouts are relatively "busy"/"complicated" situations, so I suspect many drivers have less attention left over to check for pedestrians.

IMO you are absolutely playing frogger with the gaps in the traffic.

To an extent… drivers tend to accelerate when leaving roundabouts which can make crossing difficult for pedestrians

Especially bad when crossings are like 30cm from the roundabout. Some are better with at least one car's length between the two.

Otherwise you either risk getting run over by a car exiting the roundabout without seeing you; or getting run over by the car that stopped, but was rear-ended by another inside the roundabout.