There is no "rule" that I'm aware of, at least at my current level of playing.
How you voice chords will depend on the context of your playing (melody on top? bottom? Roots handled by bass player? comping behind a soloist? playing modally? trying to stretch your technique and understanding with self-imposed limitations, etc).
The "side by side" method (i.e. along one string, up and down along the length of the neck) helps you visualize where the voices of the chords are moving and their intervals. As you've probably noticed, the technique can result in a lot of hand movement and can make it hard to execute changes very fast. But this limitation (or any limitation) can sometimes help you do something creative.
So don't overthink it, sometimes you want to get to a different part of the neck to add a high (or low) melody or riff. So there is no "rule". Sound, playability and ability to execute all contribute to your choices. The great thing is, Tomo demonstrates very good examples, which are perfect places to start from.
The answer, again, to all "should I" questions is: yes, learn it all, ESPECIALLY the triads for the chords... and, later, eventually, everything else, which includes, but is not limited to: all the inversions, both side-by-side and up-and-down, the 7th chords, and other extensions. Accept that it will take time and embrace the time spent on that path. It is all in the lesson videos, so just work through each one.
Take each one, break it into bite-size chunks and Don't Expect Too Fast. Tomo shows you in the most accessible genres, Blues, Jazz, Funk, neo-Soul and Pop, so you can progress in the music you prefer to start with.... but again, since there is time enough for all, learn at least something from each style, eventually.
Try to make achievable goals and celebrate each milestone. The universe of music is huge. The trick is not to be intimidated and narrow your scope from "learn everything all at once" to "learn this one new thing for now" and see where it leads.
It seems like you're hearing something in your 1-6-2-5 practice, so explore it. Ask yourself the "who what when where why" it sounds "off" to you. Can you solve it? Do later videos give you some more ideas to fix it? Just keep taking another step.
I probably don't stress this enough, and it is hard for me to practice what I preach (and I don't mean to preach, although I never seem to be able to stop typing) - but listen to actual music. Find songs that use the 1625. How do they solve the problem that you're hearing? Tomo has quite a few jams in his demonstration videos. But, it's ok to check out other songs, too. Youtube is your friend here.