Why is it that when I start recording the demo, most of the weapons disappear except for the one I'm holding? Is there a solution to this glitch?

My Half-life version is 25th Anniversary.

If you want to speedrun this game I recommend you check the resources tab and download either goldsource package or 2005 WON version which comes with BXT and it has all features such as proper demo recording. And no, bxt doesn't work on 25th anniversary version. Either way I recommend you try those other versions of hl1.