The reason most DMMs don't do this is because they use a mechanical range switch. There is too high a risk of accidentally shorting out your DUT when changing ranges. The meter should be safe in this case, but it won't necessarily survive. The reason that mA and A are separate is that the amps jack is permanently wired to the shunt resistor for minimum contact resistance. Only the sense wires go through the range switch.
If you make a meter that has a push-button interface instead of a rotary switch and can safely switch the maximum surge current, go ahead.
As for switches, you could also consider optocoupled switches as an intermediate between relays and analog switches. Like relays, they can switch high voltages without needing a high side supply, but they can be smaller, cheaper, and faster like analog switches. They use less power than normal relays, but obviously more than latching relays. They aren't without their downsides: they have non-negligible leakage current, greater on-resistance than relays, and typically lower maximum voltage and surge handling capability than relays.