BJP injected discussion of issues into Indian politics but they abandoned it after 1991.

It wasn’t BJP but Congress which made babri masjid a political issue.
The tussle over Babri Masjid is nothing new and is going on peacefully since the times of Akbar. British courts ordered a complete closure of Babri masjid for both Hindus and muslims and quarantining it completely.
After 1947, Jan Sangh tried to raise the issue of babri masjid and other ransacked temples but Jan Sangh never got any major support from Janta.
But when in 1989, Rajeev Gandhi lead a rally to babri masjid and opened up Babri Masjid gates for muslims, the issue became a national issue and BJP started getting Janta support.
I say it was congress/Rajeev gandhi who paved the path for BJP to get in mainstream politics and power.
in 1949, supreme court maintained the decision of british courts to keep babri masjid closed, but in 1986, a session court and Faisabad ordered opening up of babri masjid for muslims, yet, it was not a political issue. When Rajeev gandhi rallied towards Babri Masjid and facillitated the opening of babri Masjid gates, it suddenly became a national issue, and after death of Rajeev gandhi, BJP grope the issue and got power too.
Many people (Like TOI columnist M.J Akbar) still believe that more than BJP the then congress rulled central government under the headship of P.V NarsimhaRao was responsible for demolition. Anyways, the demolition of Babri Masjid could have been aborted, but it again was a vote bank issue. Congress played and it lost. BJP came in power.

They rejected the legal system and wanted to deal with it politically.

Agreed, Muslim vote bank is a big issue in UP, there are cities with 51% and even more muslim population. Yet its not wholly muslim groups which are responsible for politicization of the issue, the NCP was major force behind that idea and the main reason was to gain muslim vote bank. Unfortunately or Fortunately, Congress failed to gain any vote bank in UP atleast, although the politics of religion provided certain profits for congress in other parts of India.

One of the issues was the Ayodhya issue. A related issue was the Shah Bano case. In each of these, the discussion was whether the judiciary or the parliament should be involved in resolving the issue. The position of the Congress Party was inconsistent. It used the Parliament to overturn the Supreme Court ruling in the Shah Bano case. BJP was quick to point this out and wanted a quick resolution to the Ayodhya issue through political means. Rajiv Gandhi’s government criticized the BJP and thus ended up taking an inconsistent stand. BJP was willing to go to the courts too as long as the other issue was not resolved in Parliament.

A negative point for the Supreme Court was that they REFUSED to take up tha case! After the demolition of the dilapidated structure at Ayodhya, archeological evidence supported the claim that there was a temple at that site. In a new twist, members of Congress and Communist Parties changed their tune to “So what if the evidence supports Hindus?”

So, no, this is more than a legal issue. It is being decided in a manner chosen by the Muslims. They first politicized it and went on a rampage when the courts ordered the gates opened in 1986. They rejected the legal system and wanted to deal with it politically. When they failed in the political game, they rejected that too and wanted to resolve it outside the system. They lost there too and then they wanted to go to the courts. The same judiciary that had rejected the issue took it up and now admits that there is evidence for a temple. They won’t accept that either.