Many on this forum has switched to the weber, but most have not. The factory carb works pretty well when running properly, is reliable and relatively trouble free. Usually poor drivability is from a vacuum leak or mis routed vac lines, or some other component. Sort out the system first and than decided if you really want to spend $300 to replace the carb and all the system associated with it. The carb is not difficult to rebuild, kits are inexpensive ($12 to $30) and drivablity in extremely cold and very hot climates is actually better than the weber. If you live where it stays below freezing for many days at a time, the weber will still work but you will have to wait until the engine warms up before it will run properly.
The weber will give you a little more power and perhaps better throttle response, however it will not pass emissions if your state requires emissions testing.
That weber kit you listed is for the 3rd generation tercel, it uses the same size engine and the carb is almost identical to the carb in the Tercel4wd, so I would think the kit should bolt up (unless that kit was intended to replace the old throttle body fuel injection system). That engine was the 3E, vs our engine the 3AC, it was installed in a transver manner and used a 12 valve head (completely different design than ours). So it would be taking a chance to try that kit over the one that was designed for the 3ac engine.
'87 Tercel 4wd SR5 (current engine swap project)
'84 Tercel 4wd (daily driver, with on going mods)
'92 Mazda MPV 4wd (wife's daily driver)
'85 Tercel 4wd DLX auto(daughter's daily driver)
'01 Suzuki Esteem (other daughter's daily driver)