I don't think the assumption that printing money always causes inflation is correct. It is correct in the vast majority of situations an economy experiences, but not necessarily in the situation of a depression.

Let's assume a "Normal" if simple economy. It includes $100 and 100 widgets. It is at full employment and all factories are working to create those 100 widgets which therefore cost $1 each. Now let's print $100 more. The number of dollars in the economy doubled, but the amount of widgets it produces can't increase. Therefore the demand generated by that extra $100 that consumers wish to spend means they outbid each other for the widgets available using those extra dollars, and ultimately the price of a widget reaches $2, an inflation of 100% (100 widgets @ $2 ea = $200)

Now, let's keep all that the same.... But assume the economy is full of unemployed people, and of widget producing factories that are shut down and are not being utilised. In that circumstance, when you increase the supply of money by $100 there is the same increased demand for widgets from consumers seeking to spend that extra $100, but the economy can now respond. Workers are hired. Factories brought online. Let's say there is enough slack that, due to this increased demand, the economy can now hire enough workers and bring online enough factories to now produce another 100 widgets. Then the price of widgets stays at $1 because although there are now $200 there are also now 200 widgets and therefore THERE IS NO INFLATION.

Obviously, this is an extreme simplification of a real economy, but this outlines the principle behind the idea of printing money in an economy where there is high unemployment and idle capital goods. It provides the liquidity to allow the market to utilise its capacity to produce goods and we all benefit (as an economy that produces 200 widgets for the same population is richer than one that produces only 100 for the same population).

It's under this principle money printing was justified in a depression, and under this explanation how it can be done without creating inflation.

Or at least, that is the basic idea..... It gets vastly more complicated in a real economy with all sorts of other factors involved.

But, you need to remember "money printing = inflation" is only true in a situation where there is not unutilised capacity in an economy. In a situation where there is such capacity available money printing may not cause inflation and would be expected to spur use of that capacity, as can arguably be said to be occurring in recent times with the current expansion of world money supplies married to continuing low inflation.

Mor money produced only reduces the value of the current money IF it is chasing the same amount of goods. If more money being printed causes creation of an amount of goods equal to the money supplied money retains it's value AND employment goes up. This is why they printed money during the depression (whether or not you agree that this succeeded)