Not quite your question, but I recently acquired the Heaven 11 Billie MKII integrated, which is a tube preamp combined with class D amp. I've been tube rolling the last few weeks, switching preamp tubes makes quite a change. Just throwing out the tube-preamp with ss amp combo.

Sansui established this long time ago officially ending successfully  with  their project to create a S.S. design that will sound exactly as their top amplifier AU111...

Then from this came half of your sentence : some tube designer effectively created S.S. design sounding exactly as tube design... ... Effectively the tube and transformer designer Sansui going S.S. design create two amplifiers sounding exactly the same, one was tube and the other S.S. ...

The other half of your sentence about the fact that no manufacturer never said that their tubes amplifier sound like S.S. design is not exact... David Berning said so and he is one of the best tube designer in the world...he claim that his tube design sound fluid as tubes and detailed and dynamic as S.S. design...

Some tube manufacturer claimed exactly that as Berning did because S. S. design has also his qualities , faster dynamic, more details, more power etc

Then your sentence is not a very deep truth but a common place joke born from design prejudices...

The reality is any design is a trade-off master piece or not , being tubes or S. S. ; but some company succeed better than others, being S. S. or tubes...

😊

Ever notice that SS manufacturers and merchants often tell us their device ’sounds like tubes’ but no tube manufacturer sez their device sounds like SS. :-)

Probably one of the best solution could be a tube pre-amp as the Berning microzotl and a power amplifier of S.S. design which must be tube-like for this optimal pairing as any Sansui of the Au series are... ...

If i could i would have done it...But my Sansui alpha is an integrated with no separate pre-amp and amp function ... Costlier Sansui alpha are designed to be so...The Au 7700 too...

I just bought an Integrated Amp for my second system. No it doesn't sound like a tube amp , but it is detailed , has a great midrange , excellent low frequency control ....  it has it all .  Except the tube sound you are looking for.   

As mentioned previously I have a pair of Mid Mono.   It wasn't until I swapped my CJ pre for the incredible sounding Zesto Leto  that I fully realized how good they are for $2000.   

Maybe the Anthem integrated,  Great amp.   $4k plus.   Still not going to have the tone and meat on the bones you seek.  Great sounding amp though.   

Honestly , a really good piece of gear won't always allow you to tell what the topology is.  That Zesto is a good example.   It just sounded so effortless and beautiful.  It committed zero sins .   It was a transformative experience owning a preamp of that caliber .

This is my dream amplifier a S. S. one it is the top version of my Alpha 607 i :

Sansui Alpha AU-907MR

https://www.youtube.com/watch?v=Pa2Y95eQjNA