The website is allocating servos like this....

Ailerons - 2 servos - One channel
Elevator - 1 Servo - One Channel
Rudder - 1 servo - One Channel

That is the minimum required. although I would put each aileron on a separate channel, using a computer controlled radio, so I could dial in spoileron mixing for landings...

So YES a better radio is advisable...

One thing often recommended for novices that have passed their initial flights, and seem interested in keeping in the hobby.... that they purchase the BEST computer controlled multichannel radio they can afford.

Such radios can retain the settings for multiple planes. The mixing capabilities permit you to use less electronics in the plane. And one such radio will ultimately replace multiple less radios, making it a CHEAPER buy over the life the radio.

Also if you ever go to sell it, people are always interested in used high end radios, but they are NEVER interested in used low end radios.

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You can use MINI servos ( don't mistake them for MICRO servos! ).

Mini servos are about 20g in weight and have about the same amount of torque as most standard servos.
They are smaller than standard servos but larger than minis.

To be safe ALWAYS measure the dimension of the servo mount, and compare that to the dimensions posted for the servos, before you order them.