For a while I lived in an area in th US where I had a grocery store, a library, and several grocery stores within a mile. But my commute was 10 minutes by car, and almost an hour by bus. And the buses took a pretty direct route and there was a bus stop pretty close to my home. Part of it was just all the stops the bus made, and a big part of it was just waiting. The bus times weren't very reliable, so I had to show up pretty early, because if I missed the bus I would have to wait an hour for the next bus. And the bus just stopped and waited 20 minutes for a light rail to get there. And depending on the day, I might have to switch buses there.
And half a mile is still a long ways to carry groceries, especially in the heat or cold. But taking the bus required timing how long you shop so you are done in time to catch the bus that only comes once an hour.
And that doesn't even get into the hours the bus didn't run, or the inconvenience of having to carry all your groceries.
So while I think having such amenities closer to where people live, would be good, and would help. I don't think it is sufficient to solve the problems with public transit being slower.