All the plants that your landlord wants to get rid of indicate that there is a lack of specific nutrients in the top soil. The weed and feed, aka commercial fertilizer and herbicides, actually will have the opposite effect in the end. This is the common vicious circle that chemical companies make billions of dollars on. To explain this more clearly, the herbicide kills the fertility of the soil because it is poison. This makes the soil even better suited for the weeds which are nature's soil builders. Then you add petroleum based fertilizer, which also kills the soil life. And though the grass will appear a darker green after it takes up the new fertilizer, the soil life takes a dive and once the inorganic fertilizer is used up, the grass is left with less to feed on than it had before you added the chemicals. This is the vicious circle. If the shit actually worked, the chemical companies would make a tiny fraction of the money that they do, because people put that shit on year after year with the same results, more weeds, unhealthy soil and struggling grass.
The plants themselves will accumulate the needed nutrients in the topsoil over time, as nature designed them to do so. It you want a pure grass lawn, do what paul suggested, get your soil tested and amend it accordingly. But the best solution is to bring in organic topsoil or composted manure, and spread it each spring. Mow high and water only when the grass starts to turn brown. If you over-water, you leach the nutrients from the topsoil, resulting in decreased fertility. And if you mow the lawn too low and too excessively, the grasses spend all of their energy trying to regrow. This uses up their stored energy in their roots to the point where they begin to die back. But if you mow high, and do it less than four or five times a season, depending on the year, the grass should thrive.
The plants will flourish from the organic fertilizer (topsoil or
compost), but like human civilizations, when you let them flourish, all of the sudden they disappear. This means, that over time they will do their work, and replenish what is missing from the soil. Then they bow out. The amendments and the new topsoil or compost will speed up this process considerably, to the point where you can satisfy the landowner within a couple of seasons.
Or you could enlighten the landowner on all the benefits of the medicinal plants that are growing in the lawn and all that they are doing for the grass. The yarrow and dandelions are accumulating nutrients in the topsoil and both are medicinal. The clover is fixing nitrogen which the grass needs, and the moss is making the lawn even more cushy under your feet. A pure grass meadow is not found in nature, but in order to get close, you need deep, healthy topsoil. Not poison in a bag that you buy at Ace.