bismi llahi rahmani rahim
those you have mentioned are المؤلفة قلوبهم and those were both non-muslims and recent converts (whose conversion was perceived rather as an act of assimilation) who were powerful enough to cause trouble to the tiny muslim community so giving a part of Zakat to them was a containment policy to have them siding with muslims and to avoid domestic trouble as muslims (who were very few) had to face external geopolitical challenges.
Thus, the aim was not to give poor converts a financial comfort to encourage them to convert although conversion is universally encouraged in every licit mean.
The Calife Omar ceased the payment of Zakat to that category because at his time the Islamic nation was strong enough to survive any domestic trouble. What Omar has done was pure and pertinent Ijtihad.
Poor muslims or converts who came to medina and only counted on the support of the community were around 400 and belonged to ahl-assifa أهل الصفة (ie abu huraira). Their status disappeared by the time of Omar who founded "bayt al Mal" which was in charge of redistributing the money of Zakat and other ressources to needy muslims (for the first time in recorded human history).
God knows best