In general: I do not know why I should not use Ospf through internet /
There are exceptions, but generally speaking, if the traffic is untrusted, then routing is untrusted. In other words, if I'm on the untrusted side of the firewall, presumably you want to filter my traffic to make sure I don't send malicious traffic.
If that's the case, then why would you trust my routing information? I could send you bad routes, spoof networks, or blackhole traffic. OSPF assumes routing sources are trusted, but here, this is not the case.
Is it correct if I say it like this "An ospf network with intermittent
links can increase traffic every time a router sends information. This
lack of scaleability in ospf which makes it unsuitable for routing
This may have been a concern 25 years ago when bandwidth was very limited, but network speeds are so much greater now that the amount of OSPF traffic, relative to available bandwidth, is trivial.
It's also true that maintaining state tables, flooding updates and calculating shortest paths would not work at Internet scale. OSPF requires too much processing and too much memory for that. Moreover, the Internet topology does not fit very well into OSPFs area hierarchy.
But the real reason is because OSPF assumes the best path to a destination is the lowest cost (highest bandwidth) path. On the Internet between different organizations, this is not true. Paths are chosen for other reasons such as monetary cost, peering agreements, or other administrative reasons. OSPF doesn't let you control routing in that way. BGP gives you much more administrative control over what routes (prefixes) you advertise, what ones you accept from others, and how you evaluate the preferred ones.