Does using a random subdomain like the one below introduce any
security implications?
The issue isn't so much whether you're making things less secure, rather it's whether this serves any benefit and whether there's not a better alternative to serve the same purpose.
ssh.example.com
The reason people do this is because it's easier to remember a subdomain than an IP address, and so they do this albeit at the risk someone might discover the IP.
In your example it's unlikely anyone could guess the subdomain, but I doubt anyone could remember a random 64 character subdomain either - so really you might as well just remember the IP and avoid the risk altogether.
Further to that, there's better ways to provide access to administration services while still offering a similar level of convenience. For example you could have a VPN gateway on a different server/network and administer your production servers via that.
That way even if you had a vpn.example.com subdomain and they discovered and DDoS'd that, it wouldn't expose or affect your production servers. You may not be able to access the VPN until you mitigate the DDoS, but the impact on your service will be much lower.
