Why does my illuminated light switch glow more dimly with an LED bulb installed in the fixture?
It's the LED cheap obsolete switch

Many switches need power for the electronics of the switch itself.  Smart switches, dimmers, motion sensors, and lighted switches.   Power comes from 2 wires: always-hot and neutral.  The switch also needs switched-hot, so it can switch the light.  So we're talking about 3 wires in the switch box.  

However, back in the olden days, about half of all switches were wired in a "switch loop".  The switch loop only provided always-hot and switched-hot.  No neutral, since plain old switches are human-powered and don't need neutral. 

So, when things like dimmers and lighted switches started getting invented, they had a problem.  About half of installations didn't have a neutral wire.  How to power the switch??

So they exploited a weird bug/feature in incandescent style lamps.  Before they are lit, they have very low resistance and resemble a dead short.  Designers found they could power their gadgets by leaking small amounts of current through the incandescent bulb, effectively putting their device in series with the bulb.  Since this was guaranteed to work everywhere, it became THE standard way of solving that problem.  

Then in 1998 along came the CFL, using an electronic switching power supply.  This did not like small amounts of current being leaked through it. The industry needed smart, dimmer and lighted switches that sourced their power from neutral instead of leakage.  But, chicken and egg, too few switch boxes had neutral because of switch loops.  Classically at the speed of NEC, it took them until 2011 to absolutely require neutral wires in all switch boxes, and even that only applies to new work.  But at least neutral-powered products are being made.  

Anyway, you need one of those to play with any random LED.      

You may have better luck with LEDs rated as dimmable, since they have extra features designed to tolerate leakage style dimmers.  

Whose fault is this? The electrician's.  LEDs are not a surprise (especially if an LED was already installed).  In this day and age, he should have selected a neutral-powered instead of leakage-powered lighted switch, or at least warned you of the issue.