It sounds like your master gardener is merely repeating what she has heard, was told or was taught. She didn't offer any reasoning behind the practice, unless you neglected to relay that in the post (in which case, shame on you). "More weeds will come up" is not a reason. A reason would be an explanation why it is better than other methods of weed control.
There are many ways that weeds can benefit, not that those would necessarily outweigh the benefits of pulling, but there are a few reasons I see that are specific to the cutting approach. Here, I assume "cutting" means above ground level.
---- Obviously, removal of any seed producing part if the plant will help prevent the spread of said weeds.
---- Every square foot of ground will have hundreds (or thousands) of seeds just waiting for the opportune time to sprout. Pulling weeds, especially large ones, will jostle and turn over soil, bringing dormant seeds to the right conditions (most often the right depth) to begin germination. So no pulling means no germination of these dormant seeds.
---- Probably the main factor limiting weed germination is moisture availability. Seeds need free water in the soil to absorb and germinate with. Keeping the soil zone populated with roots of plants is an excellent way to suck up this free moisture quickly and prevent seed germination.
*** For the most part, this is how thick lawns prevent weeds. They prevent weed seeds from germinating by using up free soil moisture quickly, before seeds can germinate. Again for the most part, once a weed seed germinates, you've lost "the battle." ***
So no pulling weeds keeps their roots intact, keeps the soil zone with less free water, and less new germination weeds. This is a concept used by nature, wild space reclamation, permaculture and many other plant related systems.
---- Compared to bare ground, weeds do provide shade and help slow soil moisture loss when it is below the "free water" level of capacity. I am sure most of you have tackle your weeding incrementally at one time or another, and have observed this. When you return the next day to finish weeding a heavily infested garden, you discover the soil is dry where you had weeded, but the soil is moist where you still need to weed.
The real question is: are these reasons an overall benefit to the community garden. If it is like most community gardens, I'd say no. For instance, a soil zone filled with weed roots will hamper vegetable (and flower) production very significantly.
