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If the 2x4 width (about 1.5) is too small, 2x3 lumber is common. Get a bunch and screw two boards together to use three 3 inch (actually about 2.5) so that you don’t worry about the board tipping.
Since your current frame is legless, i’d probably do a decent sized frame underneath so not just supported at corners.
Board games at a local game store. Many will have a board game night.