I think YouTube would use an MSS value low enough to avoid any such MTU issues. Your connection appears to handle 1472 byte packets just fine as long as they are not fragmented.
I tried to do a tcpdump while playing a video on YouTube. After being confused by the extremely large number of TCP connections involved in watching just a single video, I took a look on the MSS values.
I found that they would send three different MSS values in their SYN packets: 1410, 1416, and 1440. Subtracting 40 bytes of IPv6 header and 20 bytes of TCP header from your 1472 MTU gives an MSS value of 1412. That means if your gateway was one of the endpoints, it would use an MTU of 1412. But more importantly, any TCP connection you open on which the MSS is 1412 or less, should be immune to the MTU problem I noticed on your connection.
Comparing that with the observed MSS values I realize that I was probably mistaken when I thought YouTube would avoid those problems. Of the MSS values I observed, only the MSS of 1410 was small enough. The MSS of 1416 and 1440 would be too large for your tunnel, thus they would require PMTU discovery.
If the tunnel is correctly configured on your gateway, then PMTU discovery may still work, and the MSS sent by your gateway will be only 1412. But hosts on your LAN know nothing about the tunnel and presumably use native IPv6 when communicating with the gateway, thus those will use an MSS of 1440.
That means the segment size agreed upon by the endpoints while setting up the TCP connections is going to be too large, and PMTU discovery will be required. Some TCP stacks will not reduce the size of the segment that triggered the packet too big message. So one end of the connection will send a 1440 byte segment, which with TCP and IPv6 headers mean a 1500 byte IPv6 packet. That will trigger a packet too big error from the tunnel endpoint. The sending host could retransmit the segment as two smaller segments, but I have observed systems that instead will fragment the segment.
If the fragmentation issue that I mentioned earlier affects not just packets for the gateway but also the computers that you try to stream videos from, then it could cause TCP connections to stall.
This is all just a hypothesis, we would need to see actual packet dumps from your network to confirm it.