...is there a gap in my thinking?
...assume the reconnect occurs long after the rocket has engaged the next stage...
I think there's the gap.
The payload fairing is typically deployed while the second stage is still firing. See this example during the first Falcon Heavy flight.
...and that the tether(s) is designed in such a way that it avoids collision with the payload/next stage...
There's a substantial length of still accelerating rocket which must pass through the deployed fairing. This cannot be hand-waved away.
Here's a SpaceX fairing test in a vacuum chamber. Note the tethers, presumably only for this test. The fairing isn't much wider than the rocket itself. The still firing second stage of the rocket must pass between the fairing halves. Tethers risk being snagged. Hinges risk the fairing not separating cleanly and impacting the second stage. Either puts the mission at risk.
With what force are the two fairings ejected from each other?
I don't know, but I do know it's pneumatic and it's a pretty light push to reduce shock on the payload.
The two halves of the fairing are fastened by mechanical latches along the fairing vertical seam. To deploy the fairing, a high-pressure helium circuit releases the latches, and four pneumatic pushers facilitate positive force deployment of the two halves. The use of all pneumatic separation systems provides a benign shock environment, allows acceptance and preflight testing of the actual separation system hardware, and minimizes debris created during the separation event.
Source: Falcon 9 Launch Vehicle
Payload User’s Guide rev 2 section 2.3
What is the weight of each fairing pieces.
I don't know, but I do know its dimensions and what it's made of.
The fairing is 13.1 meters (43 feet) high and 5.2 meters (17 feet) wide. It consists of an aluminum honeycomb core with carbon-fiber face sheets fabricated in two half-shells.
The Falcon 9 Launch Vehicle Payload User's Guide says...
The SpaceX fairing is 5.2m (17.2 ft) in outer diameter and 13.2 m (43.5 ft) high overall. Fairing structures and dynamics result in a payload dynamic envelope with a maximum diameter of 4.6 m (15.1 ft) and a maximum height of 11m (36.1 ft).
I figure, if the fairings are jettisoned and there is a tether(s) between the two objects with enough strength to resist breaking it could reconnect...
A payload fairing is designed to be as light as possible. Any weight you can shave off the fairing is extra payload you can carry. It's also designed to be as reliable as possible, if the fairing fails to deploy or strikes the payload the mission is a wash. Making the fairing as simple as possible helps with both weight and reliability.
If that were solved, the tethers themselves would add weight. The tethers would have to be wound back in requiring a winding mechanism, adding weight, complexity. The mechanism would require an independent power source adding weight and complexity. You'd have to reliably realign the latches and latch them securely, adding weight and complexity. If you miss the alignment, you risk being unable to correct that.
When you tug on the tethers, either deliberately or simply because the fairings drift too far away, in atmosphere there's air drag to slow them down. In the near-vacuum when they've separated there's nothing to prevent them from possibly crashing into each other.
Rejoining them safely and reliably would be tricky.
Instead SpaceX uses attitude-control thrusters and a steerable parachute. The thrusters are probably cold-gas or monopropellant for simplicity and reliability. The thruster helps keep the fairing's orientation on reentry. The steerable parachute slows it and keeps it on course for landing.
Weight and complexity is why SpaceX doesn't land rockets with parachutes. However, it does use them on the fairing. Presumably the much lighter fairing requires a much lighter parachute.