Landlords do not set market rents. Renters and their employers do. Tax changes do not raise or lower rents. They make it more or less likely that new housing will be built, which affects future rents.
Land value taxes are so attractive because they avoid changing the incentives to build.
Of course landlords set the rent they're going to charge for their property. They're free to do so, within certain bounds established by the market and based upon the value of the property they are renting. A land value tax, which would be applied to ALL landlords in the area, increases ALL of their costs at the same time, which forces ALL of them to raise their rents at the same time. It would shift the entire market up, raising the cost of living for everyone who lives there.
Some employers might try to raise compensation accordingly, but they're going to have higher rental costs too since most employers have to rent their office space. So it's more likely that they'd be forced to reduce the size of their workforce to reduce costs and spread compensation among fewer, somewhat better-paid employees.
The impact on landlords would be immediate, and the impact on renters would be as soon as they get or renew a lease. Employers and employees would see an impact somewhat later, since it takes a while before a business is forced to downsize. Any new housing would be a long way off, well after all of the negative impacts have occurred. New housing may never come, if businesses and workers are forced out of the area to someplace that doesn't have an LVT.
Your premise doesn’t make sense. A land value tax would not “increase all of their costs at the same time”, unless the tax was implemented in such a way. More realistic would be that the tax would be revenue neutral, such that taxes would increase per unit dwelling on land with poor utilization, but decrease on land with better utilization (i.e denser). This would be reflected in rents and would encourage people to move to denser neighborhoods because the rents would be cheaper.
Land value taxes are so attractive because they avoid changing the incentives to build.