All boards have a place for the system speaker to be connected. On that board it SHOULD be located here. Keep in mind, the system speaker is only for error codes, it's not for use as an actual speaker, if you're weren't already aware of that.
So, looks like you're right, but it's the only motherboard from the last at least 15-20 years that I've seen that didn't have a system speaker header. But, looking at the specifications in the manual and looking closely at the board, there is not. So there is no way to connect one and the board likely has no way to communicate errors even if you did. I see no mention of any other method in the manual, at all. Very strange, but it is what it is.
I'm not sure even they can tell you anything without throwing parts at it since they will not have any way to identify the cause either without simply replacing things until something fixes the problem. Given the original low quality of that board, I'd suggest simply replacing it with a better model or moving to a newer platform might be wiser than spending money on a repair shop.
My plan is going there with just the board, cpu and a stick of ram and "borrow" the other parts. If its more than just the ram problem i will research for a new board, while giving this board a long soapy bath.