As I don't have time to read through the thread first, I am just going to offer my suggestion from prior editions (mixed with 5E concepts now):
When characters go to 0 HP, they have to make a CON save DC = 10 + damage overflow from hit (i.e. how far negative they went). If they make it, they still fall prone but remain conscious. Their speed is halved (optional), but they can get up and move, bind their own wounds, use an item, etc. Once they stabilize, they don't make death saves unless they act. I don't allow actions such as attacking, casting a spell, etc. unless they choose to automatically fail a death save, in which case another CON save is required to remain conscious. This way, if the damage doesn't take them too far negative, they have a chance to remain conscious and stabilize themselves.
For example, if a character has 5 HP and gets hit for 9, they have a "overflow" of 4 points of damage (since only 5 was needed to take them to 0 HP). The DC is 14 (10 + 4 overflow) for the CON Save. If they fail, continue RAW. If they succeed, they get move at half speed and can get up on their next action (since they fall prone), and act to stabilize themselves. Any "rigorous" action (your call) means an automatic death save failure and another CON Save to remain conscious.