• Your router can send out speeds of 600-1500 Mbps. The weak link isn't your router.
• Your modem can handle 343 Mbps. The weak link isn't your modem.
• A CAT 5E ethernet cable, if that's what you currently have, can handle 1000 Mbps, it is not the weak link, but if its frayed you should get a new one (doesn't need to be CAT6 as you won't reach that capacity anyway). *Your PS4 has an 802.11n wifi network adapter. It's max speed is 600 Mbps. It is not the weak link.
• HD video requires 5+ Mbps speed from your internet, with 4k needing 15+ Mbps.
• Online gaming varies but you will likely need 3-6 Mbps, but latency is king and if you have anything else coming through your internet service it is going to suffer.
• If you do a speed test you won't have a full 15 Mbps internet coming in, probably more like 10-12 Mbps.

The weak link is your internet speed. All of you devices/connections can handle speeds you aren't going to come near with a 15 Mbps connection. You simply do not have enough throughput to play online games while someone else hammers on the connection watching HD video. I have a 100 Mbps connection and my online gaming experience is diminished if someone is streaming video.