Hmm, before some moderator closes this q as being a matter of opinion, not substance ...
I think you need to consider whether the first two manifestos (to which TTM was a response) had any impact? I'd say little or no: there were a few OODB ideas that made it into the SQL standard (but were little implemented). There are a few niche commercial products that claim to be OODBMSs. But mostly the industry has stuck with SQL tables; and applications or development tools provide an ORM layer (Object-Relational Mapping).
A comparable question: have 'Codd's 12 Rules' had a significant impact on the DBMS world? I'd say not much, even though most industry practitioners would at least be aware of them.
Hugh Darwen, by the way, is not principally a "DB theory researcher". He was part of the UK development team for an early IBM relational system Business System 12 -- which arguably was more relational than SQL has ever achieved [**]; and he was IBM's representative on the SQL standard committee for many years. He only lately moved into academe. His first draft of TTM was a visceral reaction against the fulminations of the SQL committee, written while he was at one of their plenaries.
Chris Date, IMO, is not much of a researcher either: he's more of an educator and writer of textbooks. I'm unconvinced he's added much to the research rigour of the TTM document.
Apart from that, your q seems to me to show many confusions.
If you read TTM (and the supporting explanations), I see no intent to "harmonise" with OOP. Note that an OODBMS is not aimed purely at OOP's: it's aiming to support notions of encapsulation of data. TTM's claim is that a Truly Relational DBMS already includes sufficient means to encapsulate data -- including under the Liskov/Wing Substitution Principle; and that SQL has bastardised the applicability of the Relational Model. Separation of concerns is principally to be achieved through schema design; we don't need perversion or subsumption of the Relational Model.
What has support for user-defined types got to do with it? Any modern language in any paradigm has those. The usual catchcry is 'types are orthogonal to the model'.
A large part of the TTM Prescriptions and Proscriptions are taken up not with opposing OODBMSs, but opposing SQL. Has TTM had a "significant impact" on SQL? I'd say no: SQL has continued to get worse and worse. Each revision of the standard has retained all the old awfulness (due to the shackle of backwards compatibility); has added new awfulness; and has multiplied up the effective awfulness in trying to support multiple 'features' that are in tension.
There are a few products that have attempted to implement TTM's ideas -- see "Projects" linked from the TTM website. I'd describe them as 'hobby products' developed and supported by individuals -- individuals who are rather cranky, strongly opinionated and difficult to deal with IMO. I would never recommend anybody use these products. (And for that reason, I'm not going to suggest you go ask your question at the TTM forum. You have been warned.)
[**] The more I discover about BS12, the more impressed I am by how much it got right even when the Relational Model was very far from mature. How come it was killed off but SQL continued? IBM internal politics, plus commercial pressure: Larry Ellison stole some significant brainpower from the IBM System/R research team (who arguably never understood what Codd was talking about); they re-implemented at Oracle what had been only a prototype at IBM; then commercial pressure resulted in a race to get 'something' to market, and never mind the quality. Mike Stonebraker/Ingres did originally try harder to follow the model, but eventually had to cave to commercial pressure.