Around 1960 a report said that Glasgow had lower car ownership than any other city in the UK. Instead of this prompting major investment in rail/metro/segregated bus lanes etc, the response of the City Fathers was to persuade the Scottish Office to fund the bulldozing of the M8 through the West End (there was meant to be a complete motorway box, but the money ran out and anyway the public were opposed ). The low car ownership was seen as an embarrassment, reflecting on the declining economy.
As for getting between parts of Glasgow in 15 minutes, it takes me that just to get in from Anniesland to the M8. Edinburgh buses are a nightmare at present because of the tramworks and the refurbishment of the North Bridge. But otherwise they are generally recognised as excellent and keep winning national awards. Because they are still largely owned by the City, they have been integrated with planning developments, with many routes terminating at major destinations - the Gyle centre, Ocean Terminal and now the Infirmary. In normal times it's astonishing how busy buses are even outside the peaks, and there aren't just Night buses, there are buses on Christmas Day and New Year! And there are a lot of orbital routes which avoid the centre and seem very successful.
At the same time as Glasgow was being given its motorway, Edinburgh, if I remember right, had to fight to build its bypass which transformed getting from one side to the other, until it became grossly overcrowded! Can you believe that it was claimed that the city didn't need a bypass because everyone was going to the centre?