Are Roma flush with cash? If not, hard to see how Mourinho helps them close the gap on the Milan clubs, Juve and Napoli.
They're a sell to buy club. They're owned by billionaire American Dan Friedkin but they can't really shop in the same markets as those teams above. Their plan was to operate a kind of Sevilla/Porto model when Pallotta owned them, and try to balance bringing in prospects cheaply, developing and selling at a fat profit whilst still performing well domestically and in Europe. It did not go particularly well on the actual performance side of things, though they did bring in players like Salah, Lamela and Alisson in during this period and sell for good profits.
They bought Monchi in from Sevilla, a true transfer guru to this end, even paying a €5m buyout clause for him (relatively unheard of to pay for a Sporting Director) but the project never had the successes of Sevilla (Roma fans a lot less content with chasing the regular pack domestically and not making a mark on the Champions League). Despite some decent signings. He didn't see out his contract and was back at Sevilla within 3 seasons, where he's managed to rekindle the old magic.
For context, Monchi made €250m profit on player sales in his first stint at Sevilla, but he spent more in two seasons at Roma than in 5 previously with Sevilla, buying more expensive players to start with. It seems like he's brilliant at spotting mostly unheard of players and recognising they can become stars, but he was less effective at buying established players and overseeing them become world superstars. Every season he was at Roma, they did worse than the season before, fuelled by selling their best players and not replacing them adequately. He also strangely got a hard-on for the Eredivisie whilst at Roma, something he'd never really looked at while at Sevilla, so he wasn't shopping in areas he had expertise in. Schick, Pastore, Defrel, Karsdorp, Under and Santon were all underwhelming signings.
So, having answered a question you didn't ask, no they don't spend great amounts compared to the teams you suggested, no they're not flush even though their majority owner is personally wealthy
It's kinda like John Henry, he's worth personally around £3bn but doesn't use his own cash on the club.