More out of curiosity than anything else, but, what is your antenna design & construction experience like?
i'm not an expert, my experience is like only few weeks on a wireless door bell project... i havent gotten back into my rc plane project, all parts, transmitters receivers, BLDC/ESC drivers, motors and propellers bought few years ago. i built a 3d printed plane that crashed few seconds on its first flight test
took many weeks to build it just to see it disintegrate in split second, and then i dont have a chance to build another one again. my 5.8GHz wireless video got broken (my conclusion due to mismatched/crap/hunglow antenna) and bought the replacement front end amplifier, but never gotten to solder/repair it back, those experience is what drove me to find few devices up to 6GHz BW. enough mumbling, what i can really certain, from short experience and advices from people in the internet is... you cant design an antenna effectively without a measurement device, and those are like VSWR, VNA or spectrum analyzer things like that. without those you can only do the field test, ie actually playing with it and see how far can it go. try to build one and hoping it can do well is like walking in the dark with many deep holes everywhere, the chances that you can do it right in the first try is like 1 in a million. ymmv.
That "dicky wire" antenna is actually a center fed vertical dipole with very similar performance to the one below it - just in a more compact, easy to use form - one 1/4 λ element is the exposed center conductor of the coax, the other 1/4 λ element is the tubular sleeve that's over the coax - together they form a 1/2 λ dipole - fed at the center point. This type of antenna is very common on indoor WiFi routers
iirc, my vna didnt tell me that, at least on hunglow antenna similar to whats linked by OP, but i can never know what exactly he will get, maybe the one he will get is built with specifications as you've mentioned i'm not sure.. simply put, if the performance is the same, there is no need for me to built my 120 degree dipole. i can snap a picture i have one lying around but i dont think its worth it if you never ask, i have few fancy GHz antennas incl mushroom here bought from China and i can tell you they are all crap in term of what a VNA can read. cheers.
edit: and i forgot that i also built a small yagi antenna for a 2.4GHz broadband dongle that kind of worked with hunglow coax cable long before i bought my VNA, using plan and formula guidance in the net, i cant remember if i ever put it on the VNA test, it was dismantled when i upgraded my ISP to telephone line (now is fiber)