"a site this size can't cope with high traffic"
High traffic is not the problem. Automated tools and bots are the problem.
Some videos generate money for the person posting the video - so, some posters try to use automated tools to repeatedly "watch" their videos. The captcha is an attempt at stopping those automated tools from "watching" videos.
Before you say that you are not using an automated tool to do that and you are not infected with any bots, you need to understand how you are connected to the internet, how Google (YouTube) tracks networks (and what they consider a network), how proxies work (if you are using one) and where various NATs may be between you and YouTube (e.g. a cluster of dwellings behind a NAT device).
If you can, I would try and get a new IP from your ISP and see if that solves the problem for you. Unfortunately, that just pushes the problem off onto some other unlucky person. If Google (YouTube) sees too many IPs from a give ISP using these automated tools, they will turn on the captcha block for an entire range of IP addresses.