The border encounters did not skyrocket under Bush. They were lower than Clinton's last years. The numbers were good under Obama, but Obama's last three years were a bit higher than Trump's first two years.
We know the border crisis really got going in 2019 with the surge in non-Mexican migrants at the Southern border. We know COVID alleviated the crisis in 2020, but we can't negatively judge a President for a good result. We also can't judge a President for what he would do after he is out of the office.
The Biden Administration got hit hard with the border crisis ramping up his first year. As we saw the press beat up on Trump for alleged failings in dealing with the rush of migrants, now Biden is taking his turn catching a beating for perceived failings in addressing the crisis.
It's my opinion, that the Biden Adminsitration hasn't walked the talk in the way the Trump Administration did. It's always peculiar in politics that in the course of rhetoric, people will beat up a leader for his policies. Then those same people will turn around and defend their leadership choice with statements that he's not to blame he's following the course of his predecessor.
Doesn't matter if Biden broke it, he owns it. His party also had control of both the House and the Senate.