So based on the million seller list Nintendo sold about ~85M units of first party software this FY (in reality probably more due to titles not on the million seller list, but probably not terribly much more).
Which would be about 50% of the total software sales, which is nowhere near the reported 82.8% "proportion of first party software sales". So that confirms my suspicion that that number, which people often cite as proof that third parties don't sell on Switch, is talking purely about revenue and not units.
Which is obviously gonna be that high when most Nintendo titles are sold at full price, and also because Nintendo receives only 30% of the revenue of third party titles, whereas they receive 100% of the revenue of their titles.
Hmm, the math works out: 0.5/(0.5 + 0.3*0.5) * 100% ~= 77%, which is very close to the indicated number of 82.8% (I'm using 0.5 the proportion of first party software sales, which is obviously not the exact value). Very interesting. That could mean that third parties are contributing 70M-80M to the yearly software sales.
If we assume that that is true, then we can work out third party lifetime sales on the platform. Let's give it a try:
FY2019 {01-04-2019 through 31-03-2020}: 69.03M (proportion 82.8%, total 168.72M)
FY2018 {01-04-2018 through 31-03-2019}: 46.46M (proportion 83.8%, total 118.55M)
FY2017 {01-04-2017 through 31-03-2018}: 24.20M (proportion 85.3%, total 63.51M)
I have chosen not to include FY2016 because Switch wasn't around for most of it, so the numbers doesn't tell us a great deal. What this tells us is that out of the 350.78M units shipped between April 1st 2017 and March 31st 2020, 139.69M are third party software. That doesn't sound too bad considering basically all AAA games skip the system.
NOTE: All of this is assuming the proportion number is based on revenue. If it's not, then you can discard this. There is some messing up going on with the 3DS and WiiU's residual software sales, but that shouldn't have a big impact.
Mathematics used to arrive there:
Let x and y be the amount of first party and third party software, respectively. N is the total software shipped, as reported in the earnings releases. Let P be the reported proportion of first game sales (in revenue terms for Nintendo). Then, the following system of equations must hold:
{ x/(x+0.3*y) = P,
{ x + y = N
Now, to compute x (and y as a consequence):
y = (x-x*P) / (0.3*P)
N - x = (x-x*p) / (0.3*P)
N = (x - 0.7*x*P) / (0.3*P)
x(1 - 0.7*P) = 0.3*N*P
x = 0.3*N*P / (1 - 0.7*P)
Using this formula, you can directly compute x, since N and P are known for each fiscal year. y is calculated simply by using y = N - x.