In the cities we do invest in that too. Also cycling is superior (faster, cheaper, more convenient) to public transport for short enough trips.
Also public transport and cycling are not opposites of each other. Here in Helsinki a lot of people cycle to the train or metro station and then take that to the city for their commute. You can also take your bike into the train or metro here if that works better for you.
People also drive their cars to the metro/train station and do the same. Parking can be really expensive (and hard to find) in the center if you happen to work there. For example my work place (a bit over 100 person IT company office) we have a total of 2 car parking spots for the whole company that you reserve in a google sheet when you need it.
If you're going to invest in infrastructure, I would think that investments in public transit will be more palatable to the taxpayers.