But here's where you are in error linking all non catechical schools with modern state schools where RE is taught.
Not really - sure things have moved on in the last 150 years, but the foundations established for the secular schools in the 1870 act remain clearly visible today in our non-faith schools. Notably that RE must be on the curriculum, should be taught without favouring one denomination over another and that parents have the right to remove their children.
Where was the evidence that founders of secular schools wanted these ''bible-reading'' schools abolished...
Again you are misunderstanding things - the 'bible reading' schools was a derogatory term used by the church establishment and the National Educational Union to describe the secular schools - in effect that they merely 'read the bible' rather than were required to worship and be instructed into a particular denomination. So, of course the secularists didn't want to abolish the 'bible reading' schools, as these were, in fact, the secular schools that they had established.
... or even church schools abolished?
As far as I am aware they didn't want the church school abolished, however they did campaign against them receiving public money. This all came to a head with the 1902 Education Act which established the system of funding of church schools which isn't a mile away from what we have today - at that time state funding became available for secular teaching in the church schools, but not for religious instruction nor for maintenance of the buildings.
The reality is that the relation of the churches towards eduction and the state has followed a familiar cycle over the past 150 years, namely.
1. We want to be in control of education, we don't want the state involved - we oppose anyone else being involved (1870)
2. We've gone bust, we want the state to bail us out (1902)
3. Sure we have state funding but we must be private, we don't want the state involved (1902-1944)
4. We've gone bust, we want the state to bail us out (1944)
5. Sure we have state funding and are state schools, we don't want the state involved in any decision making (1944-1959)
6. We've gone bust, we can't afford to maintain our schools, we want the state to bail us out
7. Repeat 6 several times as the amount of funding churches are required to input into maintaining their schools (despite them owning the assets) has dwindled to effectively no different to any other state school (who don't own the assets).
So in effect cycles of the churches telling the state to F-off out of their business and then going cap in hand to the state when they'd run out of money.