Handling performance: most important chassis bracing/reinforcement - per application?

You know I think too many people buy stuff that's not needed.

My opinon on the chassis is that an important factor is the base body, and secondly how well it was built. A convertible is obvious, but some of the others are not. @CPDave has found that his goldfish (66 Barracuda) is pretty darn good as is - even autocrossing on R-comps. The transmission tunnel to cross braces in my 67 coupe have had to been repaired several times. (and I added basic frame connectors before any autocrossing or drag racing).

Most know that Dodge and Plymouth added fender braces, and while they may or may not effect handling, they didn't do that just to spend money. Post cars ought to be stiffer than hard tops (generally) and some time around '70 cars got braces inside the doors.

TIRES
This is the biggest variable. On street tires none of this is going to make much difference.
Depending on the chassis may notice some difference with torque boxes and frame connectors. I won't guarentee it though.

First check for quality and condition of the factory welds. On my coupe, the torsion bar anchors and sub frames were all good, but some of the body and in particular the cross braces behind the seat were poorly fitted and therefore poorly welded.

Drag strip: If using drag slicks and the car is rising on launch (as it should) frame connectors would help. 4 point may help a bit more and obviously bracing to the front would be most effective.
I worked with a guy that would leave his '67 GTX on strategically placed jack stands ever week to get the car to settle back. This way he didn't permanently wrinkle the quarters. No connectors, no bars - Yes 10" slicks, pinion snubber and spring clamps setup for racing.

Autocross. As much as the rules allow and you are willing to tolerate. Street based classes are fairly restrictive on how much can be welded up. No seam welding (unless factory), only three attachment points allowed on frame connectors and only two may be welded (Street Prepared based classes). Serious C Prepared effort, running the max tire allowances, 6 point is going to be a smart starting point.

Racing. Again its the rules and the tires. Running vintage classes restricted to Street TDs, Blue Streaks etc is going to be different than a GT or open class.

Track days. Let others speak to that. Other than convertables, PDX and other non-timed events are generally pretty relaxed. Some of this will be driver tolerance. Dr. Bob Reed (former road racer not th edrag racer) found his a-bodies very soft compared to the mazdas he had been driving as a pro. But Bob can drive a car much closer to its limits on a track than most of us. The better you get, the more speed and grip and loads you will be able to sustain. Some bracing is going to help the car even if you can't feel it.

Flex or lack of feel has never been an issued raised by national level drivers when they've been instructing and driving my car. It's just not that critical to setting up a good car. I've had replacing the current frame connectors on my to do list for several years. I'm sure it will help, but hasn't been a top priority.