I believe you got two issues mixed up, the old "first field white or red" debate on the Croatian coat of arms and the general unease about using Croatian symbols. But let's go from the top:

Nationalism was always the greatest issue in Yugoslavia and when the situation in the country started firing up in the early 90s, the individual republics (primarily Slovenia and Croatia) first got wider autonomy from the central party and then had democratic elections and the declarations of independence were considered as part breakup/evolution from Yugoslavia and part renewal of earlier independence.

The preamble for the Croatian constitution of 1990 listed a dozen historical basis for independence and notably one of them was the Socialist Republic of Croatia (aka Yugoslavia) and the WWII era Antifascist council's decisions opposite the Ustaše, which still stands today.

In light of all the historical reasons, many features of the "new" country were derived from old symbols and the coat of arms wasn't an exception. Here we come to what I believe is your confusion.

a) There's still a big debate about the coat of arms and to sum it up some people believe the first field should be white and others believe it should be red. As you can see here historically it's been both since the 15th century and the difference is mainly political, with the right wing thinking it should be white and local Serbs thinking it's too close to the Ustaše you mentioned. Note that it was used in Yugoslavia as well so as a symbol it wasn't controversial until politics came in.

b) Franjo Tuđman believed that after nazism and communism the "new" Croatia could finally be united and to appease the right wing of the ruling party he used dubious iconography and alluded to the old Nazi puppet regime. This naturally frightened the local Serbs and Milošević's propaganda machine used it to great effect to mobilize them and use for his own purpose. It's still a very heated subject in both Croatia and Serbia with Tuđman himself expressing regret for using such terms.

Now that this is cleared up:

Why did Tuđman and his cohorts insist on this symbol?

It was a legitimate symbol of the country, but Tuđman used radical nationalism as a tool to unite the country before what he saw as an inevitable war. The jury is still out on his role in starting it, but he certainly didn't stop it.

Did Tuđman et al knowingly provoke the Serbs within their borders?

Yes, but more with speeches and laws than symbols like the coat of arms.

Why is this symbol still used today on the Croatian national flag?

Again, it's a legitimate symbol of a country so the Nazi comparison doesn't stand.

Furthermore, what did Josip Broz Tito think of this symbol?

I haven't found any information but since nationalism was suppressed in Yugoslavia, all national symbols were sparsely used. In the end, the red-white checkboard was the official coat of arms of socialist Croatia and as such was used in formal occasions, much like the corresponding Serbian national coat of arms.

BONUS

Tuđman's idea of a national reconciliation was unsuccessful later but managed to unite the country during the war. He was really in a specific position to do that since he had both left and right wing credentials as a Partisan in WWII and Jugoslav Army general and later Croatian dissident.

The full coat of arms used today, with the crown, is considered by professionals to be kitsch and historically incorrect. But Tuđman liked the grandiosity.