Great discussion. Thank you everyone for chiming in. Keep the good ideas rolling!
I realize the limitations will make the goal an uphill battle and that there's no perfect solution. But I refuse to believe that we can't do better while still on a budget. I may be wrong on this, but my personal opinion is the reason that tipping buckets are the standard is not because "it's the most accurate technology that doesn't cost thousands of dollars to produce"....But moreso because they're the cheapest method to produce (more profit for the companies), are convenient, simple design, use the least power and are "accurate enough" for the average hobbyist to not know the difference. I believe if we really want better and are willing to sacrifice a few of the above parameters, it could be done...and without breaking the bank. If I'm wrong on that, I'll have to learn it the hard way through failure.

Heck, I have a PTZ camera that I use to zoom in on my stratus gauge and I can easily read it remotely through that. With enough time and programming, could probably train AI to read it through timed snapshots. Would still need an auto-emptying method and would need proper lighting and image contrast, but it's doable.
And probably the bigger issue at least for residential setups, is finding a location in your yard that won’t be affected by your house, trees, nearby homes, and finding a location for the gauge that won’t be affected by periodic wind bursts during significant rain events.
Very true, but this is an unavoidable limitation, even when using a manual gauge. You've also got issues of wind blowing rain at an angle limiting your effective catch diameter. Trigonometry tells us that a 4 inch diameter gauge opening only has a 4 inch diameter effective catch area when the rain comes straight down. When it's blowing at an angle, the effective catch area lowers (stronger the wind, the bigger the angle and the less effective size of your catch area)...Ideally, the top of the collection area of a gauge needs to swivel and tilt at an angle with the wind, but then you'd need some sort of a vacuum motor to hold the rain in when your gauge leans over. Something to consider at a later date..My current goal, however, is simply to match the accuracy of a manual gauge, rather than beat the accuracy of a manual gauge. So I'm not too worried about limitations that would equally affect a manual gauge put in the same location.
What about a method where a phototransistor is used to count drops from a calibrated orifice as they interrupt the light from an LED?
It seems to me it would get all intensities accurately in volume with possibly slight delay in totalling up extreme volumes.
No moving parts. Sure to be easily coded and calibrated as direct replacement for tippers.
Maybe a bit smaller orifice than say what is in a Davis bucket. If the orifice is too big, during light rain events, water can run down the face without making a drop. Not an issue for tipper as it all collects in the little bucket.
A "pipe cleaner" or something once a day to keep orifice shiny clean would be required for accuracy.
Drip counter is definitely an interesting idea for an overall total. The lack of a rain rate ability would be a setback in real time, as well as the need to clean it every day.
In reality, if rate isn't important, the tipping buckets could be improved by having a way to collect the water and drip it into the tippers at an exact rate. Much of the tipping error occurs from lost rain during mid tip (which mid-tip time can really add up in a big downpour where a tip occurs every second or two)..With always a known constant rate, a more precise calibration could be made.
If there's a way to make the weight method work, a total value and rain rate could be calculated by the second. Even the rate would be more accurate than the rate calculated with tipping buckets. Think about it, the rate is most important during the heaviest of downpours. And that's the time when the tipping buckets are least accurate. When I watch my camera on the stratus gauge during a storm and compare it to the results on the Davis, the Davis runs way behind during the heavy rain part, but the Davis over-reports during the light rain that typically follows which allows it's daily total to somewhat catch up to the manual. During the heavy downpours, when I time the change in the manual stratus over a 1 minute interval and multiply by 60, I sometimes get a much different rate than what Davis is reporting. Much higher percentage error than what my final daily total turns out to be.
Biggest issue I'm having with my proposed design in the first post is when and how to trigger the emptying phase. Don't want your gauge to overflow, yet it would be a waste of power to have it constantly emptying when there's no rain or light rain. Would need to be a way to trigger the emptying based on conditions (say, once per day if the collected value is positive or anytime when the collected value reaches a specific amount). Emptying would be "sending voltage to the solenoid valves" for 30 seconds or so... Would need some type of automatic switch that could be controlled through a script to know when to send voltage and for how long.
One other issue to ponder is what to do about liquid equivalent of snow. Technically, the weight method works for snow without the need to melt it first. But given the proposed self-emptying method, it would still require melting.
Lots to think on!