PROBLEM SOLVED.
After doing some testing with the same server but different worlds, I realized that teleport pipes worked fine so long as we weren't playing on our main world. Considering that teleport pipes were working fine on our world in Tekkit 2.1, I focused on how the 2.1 to 3.0 upgrade process might have "corrupted" our teleport pipes. My next step was to completely remove all teleport pipes from the world, recreate the same number of them with the /give command after they were all gone, and simply replace them all by hand (assuming they were fixed). This process might make some server owners gasp, but we only have 5 players total, and the number of teleport pipes deployed in the world was only about 12. Luckily, the process worked, and upon recreating/redeploying the pipes, power and items started flowing immediately.
[For server admins] The steps I did to accomplish this were simple:
BACK UP YOUR WORLD.
Do your best to estimate the number of total teleport pipes in your world, if you care to keep things legitimate. Write these numbers down.
Stop the server.
Edit .\buildcraft\config\AdditionalPipes.cfg and DISABLE item/liquid/power teleport pipes by turning their respective "enable..." values to FALSE. Save these changes.
Restart the server. Join the server with your client, walk around for a minute, just to be thorough. Teleport pipes may appear invisible at this point, or they might not; preferably, they will be invisible, as this means for sure they are disabled—though this solution might still work even if they aren't. (I had success with both "invisible" and "visible" pipes, as I ran the solution on two copies of the world, but I can't explain why each attempt differed in this way.)
Stop the server.
Edit AdditionalPipes.cfg again, this time RE-ENABLING item/liquid/power teleport pipes by changing their respective "enable..." values back to TRUE. Save these changes.
Restart the server. Devise a test of your own to see whether teleport pipes are working — give yourself a pair of power and item teleport pipes, set up a simple pipe system, separate the transmit/receive ends by great distance using World Anchors, etc.
If teleport pipes appear fixed, /give yourself or others the number of teleport pipes that you destroyed, in order to restore their original quantities. Inform players that they'll have to redeploy all their teleport pipes.
Proudly sip a flavorful beverage of your choice.
Note: Nothing needed to be done client-side for this solution to work for us.
Now that that's out of the way, I'll say that I don't think this method is the simplest or most efficient way to solve this problem; it's simply the only way I currently know how. I did look into more efficient means of destroying/restoring teleport pipes from the world, but I never had any success. I tried using mIDasGold, but apparently it only works on true blocks — pipes, however, are not blocks, but are "tile entities." I even tried to hijack a Python script that is recommended in a Tekkit forum sticky for updating a world's tile entities for the upgrade from Tekkit 2.1 to 3.0; knowing a little about Python, I did my best, but, even as it appeared to work, it would error-out halfway through editing our world with reported memory problems.
With that said, if anyone knows of a good way to quickly edit (third-party-mod) tile entities such as these teleport pipes, such as to replace them with air en masse or otherwise, please share! It could certainly help resolve this problem for others in the future!
In the mean time, I hope this simpler, more cumbersome method helps!