Well, yes and no. There’s exceptions under “hot pursuit” which would allow warrantless entry under certain exigent circumstances. Now, that being said, in a situation where someone ran a stop sign and didn’t stop when police tried to make a traffic stop would depend on a lot of circumstances. There’s different levels of someone fleeing. In California there’s 2800.1 or 2800.2 depending on if it falls under felony evading or misdemeanor evading. If someone ran a stop sign and then led police on a dangerous chase where he ran more stop signs or red lights, drove wrong way, sped way over the speed limit in neighborhoods etc, then the police could justify warrantless entry under felony evading and being in hot pursuit. On the flip side, you’d have to ask why if someone was driving that dangerously, the police would continue and not cancel the pursuit when all you have is a traffic infraction as the reason for the original stop.
If it only fell under a misdemeanor evading charge, you’d be hard pressed to justify making a forceful entry and not just setting a perimeter and contacting a judge for a warrant. It would take longer, but you’d be covered for forcing your way in. Chances are during the way he may come to his senses anyway and come outside, either by his own choice or his family talking sense into him. Either way, they’d be fine to go in at that point.
As for the beating, you’d be hard pressed to find something to justify that level of beating. If someone is fighting that hard that they needed to be beaten that badly, then you’d be able to transition to other force options that would probably allow some level of compliance to be able to get him into handcuffs. You don’t continue to fight and beat someone just because you can. At some point people have a natural instinct to cross from just resisting arrest by not giving you their hands, to fighting back and defending themselves if they’re being punched in the face and beaten. Unfortunately if that happens, it just leads to more of the same.