My issue with the Medusa's iconic ability is as a PC ability in 5th edition the rules would likely break down into save of 8+ Proficiency + Cha Mod, on a race likely to have a +2 Cha mod. So you would end up with Cha 20 Medusa PC with a DC 15(8+2+5) petrifying gaze attack.
A PC medusa's save DC would likely break down as you have suggested. However, it all depends on how it's done. Recall that I mentioned
On top of that, the other likely racial ability would be Multi Attack from the snakes.
Multiattack doesn't really come FROM the snakes. Multiattack is an ability to increase damage output so the monster can be a viable threat in a fight without relying solely on petrification to take out her foes. Keeping multiattack is also a symptom of bad monster-to-PC translation. Multiattack is far more fitting for a fighter-ish character than for a rogue or wizard or cleric. I would expect multiattack to be stripped away for a playable version of the medusa. The snake hair might be kept as a bonus action attack, sort of like two-weapon fighting, but either way I'd expect the poison damage to be toned-down given that a starting player version will have 1 HD while the MM version has 17.
"Oh sweet, you averted your eyes ? I get advanatage on all 3 of my attacks"
"Oh sweet, you didn't avert your eyes and your restrained? I get advanatage on all 3 of my attacks"
"Oh sweet, you didn't avert your eyes and your petrified? I attack the other guy"
That scenario might be fun for you as a player, but it would quickly annoy other PCs and the DM as it imbalances the entire campaign.
Dragon ? Gaze Attack!
Anything that isn't blind ? Gaze Attack!
Other PC that is annoying me ? Gaze Attack!
I can sympathise with a players desire for WOTC to put out rules for things they want to play, but not everything people want is necesarily a good idea
I already addressed the "all 3 of my attacks" bit with my discussion of multiattack above, so I will focus on the petrification part here. I would fully expect the fail by 5 or more = instant petrification to be stripped from a playable version. Maybe segregate it into a racial feat. Frankly. I would also expect the save DC to either be reduced, or for enemies to have advantage on the save, until the medusa PC reaches a given level.
There are also other options. You could make the PC medusa's petrification require additional saves. Instead of two, you could go with three, with a speed reduction coming before being restrained. You could require the PC medusa to spend an action, bonus action, or reaction to force the save. You could change it so that the medusa has to keep her gaze on a target who has become restrained by her gaze to force the additional petrification saves. The spellcasting concentration mechanic seems appropriate for that.
Also, as I've previously mentioned, the gaze attack has a 30 foot limit. The dragon you mentioned, it can probably get out of range, and every chromatic dragon wyrmling I found in the MM has at least a +3 to Con saves. An optimized medusa PC would probably have a save DC of 14 or 15 (8 + 4 or 5 Cha mod +2 prof mod), which (assuming the PC stat cap of 20 still applies) would top out at an absolute high-level maximum of 19 (8 + 5 Cha mod + 6 prof mod). The wyrmlings have a 45-50% chance of success on the save.