While I know this is late, I had created a pastebin with this info to respond to this post but forgot LUL . This will go over most of what I have gathered from runs. While a little out dated, this is a good gist of what goes into 9.9 runs an what too watch for and goes into some detail of what Arahpthos basically mentioned in his post. It's broken down into different paragraphs describing the benefits and what-not of the run.
The primary focus of starting on difficulties like 3.6-3.9 is to kill the bosses MUCH faster. Entering the bosses below 9.0 is huge and can save 30 seconds rather easily with some good rng. This doesn't affect the earlier stages as much as you would thing, as an Optimal kill while starting on 5.0 would result the same time save as a optimal kill on 3.6. It just allows you to do less riskier strats or avoids having to reset immediately if things go south. Also the game takes into account how long it takes to beat the stage and will affect your score a good amount. (Damage taken is also a huge factor into it which I will explain in a paragraph below.)
Another thing to determine what difficulty to starts off with is independent on that characters run. Toon Link is a huge example of having the hardest time to finish on 9.9 even on starting at 3.9 or 4.0. I believe Corrin & Chrom are just a few of those that struggle immensely on doing the same. It just varies so much with each character its difficult to outright say what would be optimal and what would be RTA viable. Without properly testing every characters run.
(damage taken and how long it takes to beat a stage)
The run factors into how fast you can beat stages, and with how much damage is taken. The points you get from each stage can vary from 0.4-0.9 and more depending on character. I think it's above 65%-70% where you will receive 3.6 rather than 0.7-9. Or 0.5-4 if you go to the 80%-100% in percent. Even if you avoid taking damage but it takes a long time to end a stage you will usually still receive a penalty for it and most likely will get a 0.6 rather than 0.7 which does matter a lot depending on where you started. This can make runs with multi-opponent stages or huge stages really annoying to deal with. (e.g. Moray Towers) So if you start on a run with an average character and get a 0.5 your chance of finishing on 9.9 is almost impossible without everything going perfectly at the end.
(runs that can start on 3.6 and pretty consistently end on 9.9)
Heavy weights seem to have the best time reaching 9.9, and can usually start at 3.6 earliest. Some characters like Link, Young Link and even Toon Link that have Ganon as a boss, if done perfectly, no damage and you have good RNG can reach +1.4 points. Which I don't think any other boss gives you. (I think Bowser & duon can do the same but not sure.) Falcon & Puff can also start as early because of how blessed their runs are. However puff doesn't really need the decrease on the boss since she goes up against Giant DK. But with less agile AI at the start it might still grant a lot of time save.
I think that's about it, if there's anything I forgot then I can update the post or make a new pastebin with much better information, especially since a fair amount of time has passed since writing this.