Is is possible to create modular logic gates from transistors for teaching purposes?
If all you have is NPNs and resistors, you'll want to use some form of RTL. Here, the basic logic element is the NOR gate. The one-transistor gate uses fewer transistors (obviously), but the multi-transistor gate is more robust in several ways.
And there's a teaching opportunity here, to show how the NOR gate is a &quot;universal&quot; logic element — all other functions can be created by combinations of NOR gates (including the degenerate 1-input NOR gate, or inverter).
Even large systems have been built this way. The original Cray-1's logic was entirely implemented using 4- and 5-input ECL NOR gates!

One implementation strategy would be to build up individual 3-, 4- or 5-input gates on single-inline modules like these:

(source)
These are easy to plug into a breadboard socket, allowing the students to focus on the logic they're building. If you need a large number of them, have a custom PCB made.
And if you're really ambitious, you could plug them into a universal wirewrap card for larger projects and a more permanent implenentation.