I'm not a big fan. I like the idea in the normal rules that when people are dropped to 0, they know they are on a clock to die and are helpless to stop it. It creates an inbetween spot between dead and alive. One where you can be thinking "Oh crap, I'm close to dead...but I might survive."
This rule creates a situation where you have a number more options to make sure you stay alive. You can spend your action doing a Medicine check on yourself and you can stop making death saves. So, unless the enemies hit you again, you know with 95% certainty that you'll survive. As other people have said above, anyone with a bonus action healing spell will barely notice they fell below 0 hitpoints and can use being near 0 hitpoints to be an advantage to absorb damage only to heal themselves again afterward. This removes most of the tension from falling below 0, since in 95% of all cases without monsters interfering, you know that you are going to survive.
Which brings me to the other problem: Monsters now have a moving, breathing target to attack, raising the chance that you'll die fairly dramatically. Enemies don't really have a reason to attack people who are down. Especially if you follow the advice in the DMG and avoid attacking down people on purpose. But if those people are up and are attacking or even if they are at risk of attacking in a future round, you want to put them down. What makes it weird is that the people who dropped to 0 hitpoints are still kind of difficult to kill if you can't do a lot of damage in one hit. If you have 6 hitpoints, that means you can take 26 points of damage at most before you die(one hit for 11 which doesn't kill you then 3 more hits of 5 points a piece that each make you fail one death save). It gets even worse once you have some hitpoints. Once your max hitpoints are 50, you only die to hits that do 25+ damage at once. So being at 0 is a damage shield of 3 attacks, no matter how damaging they are. While simultaneously being really easy to kill by 3 kobolds with daggers.
Tactically, it seems to mean that all battles either end up with people dead or everyone alive and well. In the system in the book, it is possible for half the party to be knocked unconscious, you can make all of your death saves and become stable and the party will have to wait around for 1d4 hours to wake up since they have no healing. I find it's nice to have a middle ground between alive and dead.
Given "Damage is King" in D&D, the best tactical route while playing with this rule appears to be to attack with everything you have every time you drop to 0 hitpoints. It seems like it would encourage everyone to go out in a blaze of glory every battle. It seems that you want people to go out in a blaze of glory, so the rule may suit you.