They're out there. Search engines are just ubiquitously poisoned to the point of uselessness for common terms[1].
But searching for specific gen4 x1 10-gig chips (like RTL8127) does produce some reasonably-inexpensive options for cards with sfp+. Give it a whirl. :)
[1]: I wrote this like it's a new problem, but looking back it may be an issue as old as the hills. I distinctly remember searching for any-brand 10-meg cards with DEC Tulip chips ~30 years ago, because that was the inexpensive PCI chip that actually worked. :)
In your searching did you find any "mainstream" pcie gen4x1 SFP+ cards? I only found the H!Fiber model - which looks perfect, except that our work won't buy from that supplier.
But searching for specific gen4 x1 10-gig chips (like RTL8127) does produce some reasonably-inexpensive options for cards with sfp+. Give it a whirl. :)
[1]: I wrote this like it's a new problem, but looking back it may be an issue as old as the hills. I distinctly remember searching for any-brand 10-meg cards with DEC Tulip chips ~30 years ago, because that was the inexpensive PCI chip that actually worked. :)