Why don't all devices use this? It adds cost and complexity. Is their any other reason for not doing something?
Seriously, I'd say that there are plenty of options and implementations for this. Having two equal batteries doesn't make much sense, so often the second is used for emergency or limp-home power. For instance, your PC has a RAM retaining battery on the motherboard for when you loose power. A laptop often gives a "Low battery" warning, at which time you're welcome to reduce power however you can.
I think that your statement that 'batteries work best if they are used until they are completely drained, and then recharged.' is a little broad. This is more the case for Nickel-based (NiCd and, to a lesser extent, NiMH) chemistries. Lithium Ion cells don't suffer this memory problem. In fact, their lifetime improves if you avoid deep discharges. See this page from BatteryUniversity.com for reference.
There are a couple of options for doing more intelligent power management in your own devices.
The simplest is an ORing diode on the power supply. If all you want is a hot-swappable power supply and you have a bit of leeway for your inputs, you can connect backup battery to the anode of a diode, and connect the cathode to your main battery. When the voltage of the main battery dips to 0.7V less than your backup (Or is removed), the other battery kicks in. Be careful of leakage current into the backup battery, it might overcharge it.
Alternatively, you can use a power mux IC like the TPS110. This lets you select your input independently (or dependently, if you prefer) of the input voltages, instead of always using the higher supply.
Finally, Linear Technology incorporates what they call "PowerPath" controllers into their battery charging ICs. I've used their LTC4011 which seamlessly transitions between battery and external power, and charges the battery while running off of the external power.
Your decision to use a capacitor was the correct idea, but that one is much too small.
It's unclear what model you have, but your RPi probably consumes between 700-1000mA. However, if you calculate that 2200uF capacitor's discharge curve for 5V @ 700mA, it's only going to delay a brownout for maybe a millisecond before it drops below the RPi's minimum 4.75V. And that's assuming your battery is brand-new and fully charged.
That RS-540 motor can draw a massive amount of current for a few seconds when starting or stalled, and that capacitor will be discharged long before it's over. Some motor controllers I've seen designed for the RS-540 are rated to handle 30A for 5 seconds.
Another option is to use a battery with a larger mAh (in addition to a larger capacitor) that can handle higher current draws.
You may still want to consider a second battery so the RPi can remain functional even after the motor battery has been completely exhausted.
You want to do some research into battery backup circuits built around an SCR (A Silicon Controlled Rectifier).
The basic principle will be something like the below diagram, but incorporating two batteries rather than a battery backing up a rectified AC power source:
simulate this circuit – Schematic created using CircuitLab
I studied this circuit in a power electronics course in college. The lamp is initially illuminated by the rectified DC from the transformer, and the battery is simultaneously charged by that same DC current. If there is an interruption in the AC signal driving the transformer, the battery will discharge through R2, driving the gate of the SCR and allowing current from the battery to flow through the SCR to the lamp.
I know this isn't exactly what you're looking for, but I think it should be a good starting place for your solution. You'll just need to do some prototyping / experimentation with SCR circuits.