Great suggestion, particularly as you say for trials with a super expensive treatment relative to control.

In defense of current practice, I'd like to add that a major difficulty when running medical trials  for new therapeutics is simply recruiting patients to the trial.  Many patients enroll on the trial with the aim of getting the experimental treatment, so it's a lot easier to recruit people when your trial has a 50% or 75% chance of assignment to therapeutic arm.  

Some other important strategies that are currently hot right now:
Platform trials: One giant trial that has one control arm and maybe three to four treatment arms.  Hard to do as it requires a lot of people to work together but amazing when you pull them off (e.g. we did many of these for COVID)
Use of historical or shared control data: Why recruit as many controls if you can integrate existing data in a statistically principled, unbiased way (easier said than done of course).

From here.

I looked through some charities and tried to assess their efforts.  I ended up looking at what activities they were carrying out, how much detail they provided about their activities, the quality of any financial reporting/quality reporting processes they seemed to have and how easy it seemed to be to access financial breakdowns.  The exercise became more of a "gut feel" analysis than I would have liked but was still tremendously useful - I'm open to suggested edits/additions, and will touch on some of the further research I would have liked to have done at the end.

• PATH: a Seattle based charity - has an India Oxygen Drive.  https://www.path.org/p/india-medical-oxygen/.  Their site has good, clear financial reporting and decent rating on charity navigator.  However if you donate to them, it's not entirely clear where the money will go - their website states that "Donations to PATH are allocated to a variety of programs and projects, depending on current needs and emerging health issues." Overall rating: 7/10
• GiveIndia COVID response fund:  https://covid.giveindia.org/  I especially like that they let you choose the cause you want to donate to and have a thorough list (e.g. oxygen, food, cash).  Their website provides detail on what they plan to do (e.g. partner with local NGOs to set up oxygen generation plants and provide oxygen concentrators/cylinders) and how much each item is expected to cost.   Charity Navigator didn't rate them highly when last assessed because of a low "independent audit" score and not having enough independent board members.  They advertise a strict "due diligence framework of verifying all demand and supply channels are completed", and the financials section of their website seems to me to be transparent enough, with auditor reports and detailed breakdowns of costs.  Overall rating: 8/10.
• International Association for Human Values "Help India Breathe Again": https://www.iahv.org/in-en/donate/  Strange charity name but they seem to be doing good work, having partnered with famous spiritual leader Ravi Shankar's Art of Living Foundation. The website outlines a detailed plan for oxygen concentrators and ration kits, and they post photos of their distributions on Facebook. They make it pretty clear that: "all proceeds will be used in procuring and distributing ration kits" and elsewhere that "100% of your contributions will be utilized for the COVID relief work".  Charity Navigator gave them a decent rating (under their old rating system) and their website seems to back this up.  Their financials page also seems pretty good with independent auditor reports that have detailed breakdowns (e.g. salary vs travel expenses vs advertisement).  However the financials only seem to show up to 2018. Overall rating: 7.5/10.
• SEWA International USA:  https://www.sewausa.org/Covid-19-India-FAQs They are shipping oxygen concentrators, ventilators and other equipment.  They have specific reports on how many oxygen concentrators have been sent (400), with many more ordered.  They advertise a 4-star rating from Charity Navigator.  They publish the independent auditor reports on the financials section of their website and appear to only spend a small proportion of funds on fundraising and adminstrative expenses. Overall rating: 7.5/10
• Project HOPE: https://www.projecthope.org/project-hope-responding-to-covid-19-surge-in-india/04/2021/
  They are responding by helping to procure PPE, oxygen supplies, ICU equipment, ventilators etc.  They proudly advertise that they meet various charity accountability standards, and are ranked by Charity Navigator, BBB and others.  Their financials section publishes accounts, independent audits and annual reports and it looks like they spend less than 15% on administration/marketing.  I especially like that they run a health policy journal (but found surprisingly little estimation of the impact of their projects last year, even in their annual report).  Overall rating: 8/10