If you wanted to go this route, there are probably better control aircraft than the F-35, or a fighter for that matter. The primary benefit of the fighters is their ability to get in and out of situations quickly, and the F-35 stealthily, keeping close proximity (mitigating some electronic countermeasure threats) for command and control.
An AWACS, C-130 refueler, or a long-range bomber is probably a more credible C&C platform for the mission, setting aside the preference for proximity. Longer times airborne, more space for controls and controllers, meaning more missions can be managed simultaneously, more power (mitigating the electronic countermeasure threat by power in transmitters rather than proximity), and the potential capability of carrying many of these flying missiles within range of their target mission, reducing the range needs of the missiles themselves.
The refueler has the added bonus of literally being able to refuel them while loitering over a target area.
The downside, of course, are all the present downsides of drone weaponry (largely moral/ethical concerns). Cheaper to build, deploy, and use means a potential (I'd say probable) increase in their utilization (moral hazards here), without regard to potential casualties (rationalized locally as, "not our citizens/troops dying").
Ultimately, though, this is probably the endgame for weapon systems. It is financially the better option, it does reduce the risk to that side's citizens and soldiers (at least in the short term, long term political and economic fallout may be different), and practically anyone can already make a miniature airborne missile fleet now, though with poor range and precision at present.