Two years have passed since this question, and it seems that a lot has changed, with more options to integrate DVB-T and SAT>IP.
For example, this company makes both DVB-T/T2 to SAT>IP and dual DVB-S/S2/T/T2 to Sat>IP network tuners (Sat>IP servers):
http://www.digital-devices.eu
(e.g. see the Octopus NET V2 series)
Look for much more to come, as in September 2017 the DVB Consortium and the SAT>IP Alliance announced a formal agreement to jointly move the SAT>IP specification forward. This sounds quite exciting for anyone waiting to see the same video-over-IP standard being embraced even further for both terrestrial (DVB-T) and satellite video.
The announcement hints at a radical name change: "As DVB takes over the management of the SAT>IP protocol, it is quite possible that the naming and scope of the technology will expand to better reflect support for all broadcast delivery systems."
In TVs by Panasonic (a SAT>IP Alliance Member), I already see it named "DVB-IPTV" or "DVB-via-IP". The name "SAT>IP" is not exposed as much as it seemed to be.
Read more at:
https://www.dvb.org/resources/public/pressreleases/dvb_pr266_dvb_forms_liaison_with_sat_over_ip_alliance.pdf