You can build about anything to your liking, no matter of speed weight and or size ..
that being said:
it's easier to buy something that's for sure big enough and powerfull enough that it'll last for the years to come, and just throttle it down to the speeds you'd want.
the nice thing about electrically powered vehicles is, that you can throttle them down easily safely and reversibly (for future speed ups

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There are numerous ways to proceed to be honest, so let me just quickly mention a handfull of options:
1) you indeed
build a kart to your likings of style and size,
what you'd want to look out for is a motor that will power said vehicle (with your daughter and maybe an additional passenger as you prefere) effortlessly, which means on the generous end of output power.
And a matching controller that ideally is
programmable, then all you would want to do is to tell said controller to never exceed a certain speed (or a certain motor rpm that is)
you can also tell it to never use more than xy% of the maximum power available (which only makes sense if you have a really powerfull motor at your disposal)
And if and when confidence in your daughters abilites rise, you can reprogram the controller to speed up/power up the kart to your desires
2) you could pick an electric golf cart and use that; most come with higher end controllers which are mostly programmable as well so you can go that route without a problem,
BUT even if it's NOT a programmable controller, the throttle signal is easy to mess with,
there are two main types of throttle a hall effect sensor type (which gets a constant 5V from the controller and outputs a 0-5V signal based on pedal position)
a simple inline resistor can pull that down to what ever you feel feasible (say 3V for a 60% maximum power), and swap said resistor for a different size whenever needed.
The other type is a resistive throttle, that unfortunately most likely runs in reverse (more resistance means more power), but that again can be messed with when needed (a voltage divider can bypass the throttle 'in parts' and thus limit the maximum resistance that way)
or what I think is likely the best option:
3)
you BUY a kart that has the correct size and safety features you want,
most likely it'll be gas powered of course; you then swap the drive train against an electric one
(batteries, motor, controller and a new throttle pedal)
and again just pick a programmable controller,
Some controllers come with a governor circuit too
(basically a three pos switch that can be flipped to run slow, medium or fast by the turn of a key/flick of a switch)
So if you otherwise liked the trailmaster mid and can get it for cheap used in good condition,
there is no need to hesitate really.
But be aware of the of an electric conversion.
I'd say you'd want roughly 3kW of motor power for a kart that size (especially in hilly terrain or two passenger mode)
that means roughly 300 bucks for a motor and 200 for a controller, plus batteries plus wiring and pedal you can easily get in the realms of a 1k project I'm afraid.
'sid