The tread on walk-behind mower tyres usually consists of circumferential grooves - that is, parallel to the direction of movement of the mower when you are mowing. The purpose of the grooves is to keep the mower from slipping sideways, for example when you are mowing across a slope, or to help the mower move forward in a straight line. I have never found any actual benefit from the grooves myself, unless I am doing something a bit odd such as mowing across a slope when the grass is very wet. In your position I would pay no attention to the lack of tyre tread unless I found the mower was not keeping to the direction I wanted.
