Almost all turbojet and truboprop engine use some kind of automatic bleed valve to prevent compressor stall during starting and fast acceleration.
Compressor stall occurs when the compressor sucks too much air for the turbine wheels to handle. Excessive presure then builds aft of the compressor, and the compressor blades stall (also happens with centrifugal type compressor).
Bleeding air from an engine (air is bled just aft of the compressor) relieves that excessive pressure and the compressor can more easily "push" the sucked air behind him.
The above mentionned valve is not controlled by the pilots or anyone, and it does not use bled air to function. It opens or closes with air and/or spring pressure. It is totally separate from the valve used to bleed air for pressurization & the rest...
About how you anti-ice without hot bled air? Take a deep look at the thing you iron your clothes with. Plug it in, put an ice cube on the metallic side and see what happens (have an adult supervise you)...
About the reliability of the all electric system, well, I can very well imagine two generators or more on each engine, with separate yet interconnectable (for emergencies) circuits for different groups of systems, not unlike the very reliable electric systems of new generation planes. Plus you still have the APU, whose only task now would be to run AC
and/or DC generators, which will be enough to maintain all emergency functions, such as pressurization, heating, flight controls and instruments, a bit of lighting in case you lost everything else.
A well thought elecrtical layout would be probably safer than having a different kind of system for everything:
If you lose one bleed air on one side, you very quickly have trouble pressurizing your cabin properly, and it gets tricky when you need it for anti-icing as well. there is no way to bypass that, but there would in an electrical system: just plug it to another available generator.
Bled air and ACM's have never done a good job pressurizing, and especially cooling and renewing cabin air, frankly.
Beed air systems are very often controlled by electric valves and motors anyway!
So why not? Most everything in the cockpit has turned from mechanic and pneumatic to electric & electronic already. And apart from the mechanics, I don't see anyone complaining!
I'll do my own airline. With Blackjack. And hookers. In fact, forget the airline.