Since I have three Macs here (two PPC G4, one intel-based MacBook) that run 10.5.8, I have a few ideas.

With the Intel-based mac, there is a fair version of Firefox that should do OK with the existing flash player. In safari you can check settings to make another browser the default, so even if you don't use it after changing to another, don't get rid of it. You may find FireFox adequate, if still using a dated Safari browser as default. But I can use an older version of SeaMonkey for some of those same sites and see YouTube, with the older Adobe flash plugin.

A few youtube videos don't work if I go to some of them online at the youtube page, though. I've noticed video now appears in web pages where they did not, in my (G4 Mini) in TenFourFox current version of that I'd left the Flash plugins disabled (the default setting) but they work. While some browser technologies have allowed youtube and a few other video feeds to load without a current or supported version of Adobe Flash player plugin, because there isn't an upgrade for OS X 10.5.8 you may have to consider looking into third-party browsers that may have brought HTML 5 video support.