Did the writers of Stargate SG1 forget about the Beta gate?

No, at any point in history there are only ever one or two stargates on Earth.

From pre-history there is one gate in Antartica, emplaced by the Ancients, which we'll call Gate 1. At some point they ascended and the gate fell out of use. The Goa'uld arrived millions of years later and emplaced their own gate which we'll call Gate 2.

The SGC discovered Gate 2 in the 1940s and placed it in Cheyenne Mountain. This is the stargate that it used by the SCG in SG1: Children of the Gods. They discovered Gate 1 at the end of Season 1 (SG1: Solitudes) and placed it into storage.

The Stargate Programme used Gate 2 until it was stolen in Season 3, Episode 22 (SG1: Nemesis). The Russians recovered it and began their own covert programme. The SGC unpacked Gate 1 and used that until it was destroyed in Season 6 (SG1: Redemption) at which point they started using Gate 2 again, having first leased and then bought it from the Russians.

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To answer your specific question, the Russians didn't discover a Stargate, they recovered the SGCs original Stargate after it was stolen, held onto it for a few seasons (and incompetently used it), then demanded various considerations to return it.