Hi Balroth! Welcome tot he party and welcome to the Compendium!!!
So first off, there's no warning necessary with Chrono Cross. It is a wonderful game in it's own right; just don't look at it like a Chrono Trigger sequel and your expectations will, out of the gate, be considerably lower and will overall do you good. This is a game that takes place in a post-Chrono Trigger world. Much of it will seem foreign and un-Chrono-ish at first, but in time pieces will start to paint an image that ties it back to Chrono Trigger. Loosely. It's moreso a sidequest in the same world rather than a sequel.
As for mods and emulation, you could use a post-processor to help smooth out the lines or adjust saturation (like the video you showed), but I haven't played on PC and can't speak to it. I've only played on emulation via tablet and/or PSP.
Now... As for companions, most can be acquired on a single playthrough. There are two points in the game where you have very minor branching choices that cause a few characters to be blocked off. Without giving spoilers...
1.) There's a point in which you must choose between of of three characters to be your guide to get to a certain location. All three choices are equally fine. I tend to choose Nikki, because his side quests have the biggest impact to me from a narrative perspective. Many others choose Guile because he reminds them of Magus (haha). In all honesty, my advice is to just talk to all three guides first to see their designs and then pick the one you like the most. They each have their own unique scenario, but it's very small and you aren't missing anything by picking one over another.
2.) You also have a mid-game branch that ultimately merges back into the main storyline with no impact, so there's no worry here, either. Both branches have three alternate characters and a slightly different scenario. This branch occurs when you CHOOSE TO HELP SAVE SOMEONE OR NOT. I choose NOT to save them (despite my better judgment) simply because the characters in this path are WAY cooler. Night and day difference in how much better the characters are on this branch. No joke.
Also, the cool thing about this game is that once you beat it once, you unlock New Game + and can use a tool to auto-recruit all characters from your previous playthrough, so after three times through the game you can recruit all of the characters in the same game.
Also -- DON'T PLAY RADICAL DREAMERS FIRST. It will spoil some of the big spoilers of Chrono Cross. Chrono Cross is moreso the expanded revision of Radical Dreamers, so I recommend it first. Then you can go back and play Radical Dreamers.