Would LEDs on Tx and Rx lines cause too much voltage drop?
Today the LED industry has developed many devices called "ultra-bright LEDs". These devices produce well-sufficient light even if 0.5 mA is applied, since you don't want to lit the entire room for indication purposes. A 3-5-10 kOhm resistor appears to be sufficient, which doesn't load much any modern CMOS output pin.

For example, the green LED made by Lite-on, p/n LTST-C193TGKT-5A, rated at 2.8V and 153 mcd, produces too much light even with 5.6k resistor when connected to a 3.3V signal. So, with modern LED technology, the status display technique under this question is well working. You just need to select the right LED for the job.