Specifically, how do you remove the old primer when reloading Berdan brass?

Here in the U.S., most brass is Boxer primed. Many shooters here consider Berdan to be "non-reloadable" and throw it away. (!)

There is a tool sold in the USA to do the job (made by RCBS), but it is not very good (IMHO). It works sort of like a bottle opener.

There must be a better way to reload Berdan brass. Any insights?

Not reloading Berdan primed brass, however, I think that, at least in these nicks of the wood, the two most common methods are 1. the RCBS -tool and 2. Water pressure.

Thanks Kalju! Based on your suggestion, I did a search & found a page with instructions on how to build a hydraulic Berdan de-capper:

http://davecushman.net/hydraulic.html

Note: the parent page also shows a German-made hand tool which looks much easier to use than the RCBS tool - but it also looks quite old & apparently is no longer available so far as I can tell.

No sweat. I have been playing with a thought to begin reloading spent military brass in 7.62*39 for my Saiga (my club uses a military range and there's literally a metric shit ton of free Lapua brass just waiting to be picked up). The only problem is, that the availability of Berdan primers is somewhat occasional here.

Here too - though the Russians have recently been sending over Tula-made Berdan primers in larger batches (which sell out quickly).

There is interest though: for 20 years or so the U.S. has been literally flooded with old military bolt actions; many like the Swiss K-31 straight pull carbines & rifles, and the Swedish Mausers, are incredibly accurate but much of the surplus ammo is berdan primed.

Lapua brass is highly respected here in the U.S. Brand new commercial (Boxer) cases sell for nearly a dollar each. Probably not much interest in Berdan though - even if it is Lapua. Still, using that brass will probably result in the BEST accuracy your Saiga is capable of.