It depends on what other mods you have. An older vehicle with worn out rubber radius arm bushings and soft shocks is going to have a lot more flex without the front sway bar, but also way more body roll. A vehicle that has newer polyurethane bushings and top-of-the-line monotube shocks in good shape can still have good flex offroad without the sway bar, and will still have more body roll without the sway bar than with it, but it would be at least somewhat more controllable.
It depends on whether there is a lift kit installed as well, the stock Explorer does get an improvement in flex without the front swaybar, but the limited travel of the stock shocks and the proximity of the bump stops and the engine crossmember mean there really isn't much additional useful travel, and even if there is, it can sometimes cause damage rather than be any sort of benefit. The front sway bar actually seems designed to limit the travel somewhat to prevent damage, and keep the strain off the shocks from becoming the limiting factor. If a lift is installed, the sway bar becomes a very limiting factor to suspension travel and usually had to be modified to work with the lift.
The lack of a front sway bar can also be minimized by running dual or adjustable shocks, although not quite the same mechanical effect, but if you're running a rig with 35" tires, 9" of lift, and tons of off-road mods, a wimpy little sway bar for street use is kind of silly, and it's probably better to just adjust the suspension firmer or softer for what you'll be using it for. You don't even need a multi-thousand-dollar suspension for this, Rancho makes shocks that are externally adjustable, and can be changed from super soft to very firm with the turn of a dial.
Generally the best of both worlds can be had by making sway bar quick disconnects. Pull the pins and tie the sway bar out of the way for offroad use, put it back and reinstall the pins for on-road use.
The rear sway bar on the Explorer was pretty well designed and doesn't limit travel that much. The first gens came with a much thinner rear bar than later gens too, which lets it flex quite a bit more at the limits of rear suspension travel as well. Generally the rear bar can be removed with minimal adverse effect, though of course if the front sway bar is removed, the rear should be as well, to prevent oversteer.
