I have one edge router connected with provider via eBGP and to the core network via iBGP, running RoS 5.26.

Using routing-filters on provider neighbor I'm receiving only 0.0.0.0/0 from him.

I have to transparently send this 0.0.0.0/0 received from provider to my core via iBGP without default-originate,
just use this BGP-router as "force-self" hope.

Redistribute-other-bgp is enabled.
But still not sending absolutely nothing to core!
Counter of "Sent" packets as well as advertisement for core neighbor stays unchanged.
I have disabled all out filters for iBGP destinations.

Before this day I had very old BGP router for this iBGP and eBGP sessions running RoS 3.22 that was transparently sending default route when redistribute-other-bgp was enabled.

Or I misunderstood something and it's possible to use Default Originate "always"/"if installed" only?
Why BGP router not sending route learned from eBGP to iBGP neighbor?
Is there any restrictions on 0.0.0.0/0?

0.0.0.0/0 is always special case. And in BGP peer can advertise default route only when default-originate is enabled.