Getting a low-cost road bike equipped with handlebar shifters is as easy as clamping on a pair of suitable, cheap 6-speed thumbshifters, together with a compatible indexing rear derailer.

The shifters with steel clamps can easily be spread around the slightly-fatter alloy bars of a road bike, clamped to either side of the stem clamp.
A steel road handlebar will have the correct diameter for mountainbike thumbshifters, but alloy road bars will require a longer bolt, bent to fit and secured with a nut (so the bent bolt doesn't need to be turned).

A wider road bar is advantageous (gives more room for your hands) when using thumbshifters, especially if brake levers with auxiliary "turkey-wing" brake levers are being used.

There is no problem with a 6-speed thumbshifter being used with a 5-speed freewheel.

7-speed shifters and freewheels all have narrower spacing between gears, so won't work well with 5/6 speed freewheels with traditional "standard" spacing.