No problem with your title, you can call it whatever you want as long as it is somewhat understandable what kind of antenna it is about, but if a unskilled beginner wrongly tells me what name to use for a classic antenna as he else will become irritated at me, is a bit too hilarious.
It can be pretty hard to design your own antenna and balun without proper tools. A Yagi-Uda is as complicated as an antenna can be. You want as beginner to build a precise designed ladder without tools in a dark room.
When using a VNA for designing antennas and baluns is VNA used as a design tool during whole development phase.
It is not something that you measure final result with as a kind of design confirmation.
It do not differ much from when designing a RF stage. Much tuning and design adjustments must be done based on measurements and not seldom must design take a few steps backward if intended goal not i reached. Knowing when to take these steps backward without tools is not possible.
Transformer exist for any power level. More then 100 kW is no problem but we keep the biggest coils in steel cages so that they not tries to knock down the house due to fast changes in power or antenna load. Such wild thing have actually happen.
Simplest is to design a dipole such that it is a perfect conjugate match relative radio impedance+antenna cable.
Then is no additional balancing element needed. Different kind of baluns can ensure better dipole balance and ferrite tubes can absorb power that is reflected.
A balun can improve impedance matching by transforming an impedance for better fit of antenna relative radio and it can make it harder for reflektions to reach cable braid at cost that it also affect antenn impedance.
An optimal effective antenna is conjugate matched but various baluns can be added to for example allow wider frequency range.
A Yagi antenna, what kind of balun to select is much a question about frequency, needed frequency range and type of dipole.
If you look at a professional antenna is balun part often self-described.
Some baluns can be more complex to understand.
Bicone antennas are designed to be very wideband. Antenna element must be designed to be impedance stable over a wide range at expected impedance. Same for balun part that must have wider frequency range then what a quarter wave transformer allows for.
A such balun can be seen here:
http://www.gtemcell.com/wp-content/uploads/2012/02/DSC02335.jpgThis is a kind of balun very common in many kinds of Yagi-Uda antennas:
https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcSmRjQyXCOF7QoPZSoxkaSDjUiuHHIOTOgrudU558EXZrwim9hMSize differs with frequency. It is a cheap and simple design that is "good enough" and easy to design for conjugate match.
Also professional antennas us quarter wave transformer in several ways. This one is popular as it can be designed for very precise performance with aid of a simple tuning and because of that used as reference when measuring other antennas.:
http://www.antennamagus.com/images/Newsletter2-1/Roberts-balun-large.pngEspecially for higher frequencies can it be an advantage to design dipole and sometimes a whole Yagi-Uda antenna diretly at PCB:
https://www.scribd.com/doc/73556375/Printed-Dipole-Antenna-With-Integrated-BalunMain problem is if you want to design a high performing antenna without doing any measurements.
It will fail. Big.
I have designed antennas as professional for 30 years and I will fail if trying to design an Yagi-Uda without using tools.
A less experienced will fail bigger.
It is no absolute need for an VNA and it is in any case needed time to learn how to use it. It is very easy to measure but it is sometimes complex to measure correct.
VNA and measurement cable, when to notice or take action to avoid that they maybe will extend antenna ground plane resulting in false impedance readings, and what actions to do are such examples.
Measure with correct port forwarding is another common problem.
Without VNA do trial and error method remain. It is not nessary wrong and it is possible to reachgood result but it will take a factor 100 longer time comparing to use a VNA.
Design a simple oscillator with controlled impedance, sama as intended for antenna, and connect directly to antenna without cable or balun. Place a receiver 10 meters away in antenna directions and measure received signal level. Use a cable to extend signal reading so that you can read it while adjusting antenna.
Now tune extremly systematic. Paper and pen and tune in small steps. For impedance is it two unknown variables that should be adjusted for optimal result and it can take some time to get a feel for what is what.
When setting oscillator impedance without tools, do I use three resistors in similar fashion as when designing resistive attenuators. Allow for a 5-6 dB attenuation and your designed impedance.
That gives you a stable TX impedance without need for tool to measure impedance.
When antenna is final optimized , replace oscillator with intended balun and connect oscillator either directly at balun, now adjusted for assumed radio impedance+cable impedance, and tune the balun, if it is a tunable version or else can it be needed to continue to tune at the antenna.
If none of above seems doable for you, you want to design something fast and simple, with some directive gain, you accept that it will be few dB loss, then can a biquad be a possible solution. It is a very uncritical antenna where 8 dBi gain is reachable with a measuring tape as the only tool and basic skill in doing nice soldering and effective connections from RF view. If VSWR meter is used can antenna height adjust impedance. Adjust by brutal bending with you fingers. Is much tuning needed is it something wrong in the design. Find out and redo.
If you want to know more about innards of the antennas and antenna-shapes, do as I do, reverse engineer every antenna within reach and investigate why thing looks as they do. Most basic antenna shapes and baluns are technically described in books like
https://www.wiley.com/en-us/Antenna+Theory%3A+Analysis+and+Design%2C+4th+Edition-p-9781118642061That book is in antenna world the only book you need.