Poetry was absolutely not expected of all men. Poetry carried different connotations for different people, and had a somewhat turbulent evolution during this specific time period (Prophet Muhammad's life). We know this because some poets were looked down upon for their poetry, like Imru al-Qais whose father said that poetry was not fit for a man on the royal path.
>(Al-Tahir Ahmad Makki, scholar says) among the northern tribes, likewise, each tribe had its chief and its poet, and the two were hardly ever the same.
I will be taking Al-Tahir Ahmad Makki's word over yours. If you want to actually read what he has to say on al-Qais, you can find it here.
Hassan ibn Thabit was a court poet in the courts of the Syrian Ghassanids who were Christian patrons and he was a court poet for Prophet Muhammad. We don't see much of pre-Islamic poetry because the scholars interested in collecting the diwans had very little interest in preserving anything considered pre-Islamic, but we know they at least existed in part because of men like Hassan who rode that line between pre-Islam and Islam (mukhadramun).
The word "professional" by itself is largely anachronistic. Most men of these time periods did not follow the rigidness of modernity. Even though there's little to no evidence of it, we can assume Hassan had other talents. But that Hassan is remembered strictly for his poetry, and that he was employed by two very different peoples for said poetic prowess, suggests that his talents as a poet were a large aspect of his life. That poets hired people to memorize, recite, and spread their poems also suggests some manner of lucrativeness to doing poetry, as do the notions of poets coming together for competitions during a time when travel was not easy.