I was in high school at the time and not exactly as eagle-eyed on politics as I am today, but the way I saw it the growing partisan divide had begun under Bill Clinton, though not because of the way he governed (coincidentally the same time the internet went mainstream, definitely a contributing factor). That four year investigation, his outright lie about the Monica Lewinsky scandal and how he handled the impeachment could have drastically altered the way we view politics today. NYT wrote what I thought was a rather interesting "What if" piece about it ("Had Bill Clinton Taken The High Road, Would The Road Be Different For Trump And Republicans?").
Bush Jr. was walking into what was already a political climate that was growing more divisive, in part fueled by the 2000 election results. The only major legislation that I remember him attempting to get bipartisan approval on was a bill about education, "No Child Left Behind," in his first two years. Some look back on his eight years as being hyper-partisan, too, but I think that's pushing it.
I liked Obama as a person and he presented himself in respectable fashion on the world stage, but I didn't think he was worthy of praise as a politician because he represented that establishment class politician that's inextricably linked to corporate America (you scratch my back, I'll scratch yours). He could have done a lot more to bridge the divide, but he didn't want to break away from his party to do it, as evidenced by his snarky remark to Eric Cantor in January 2009: "Elections have consequences, I won" after asking what the GOP wanted to see from the bailout package and being irritated by their demands. That poisoned the well right from the start, so it didn't surprise me to see the GOP react the way they did.
No real need to touch on Trump, he thrives on divisiveness, enjoys playing people against each other. He is the perfect reflection of the troll mentality so clearly on display over the internet. He won't get a second term without a wall of some sort, though. I think Ben Shapiro has it right:
The way I see it Republicans had their chance and blew it because they were overconfident they had all three branches and didn't want to compromise what's so ever. Now they lost the house and are shutting down the government for 1/5 of what they could of had.
If you can get past the fact that such Republicans no longer have a place in Trump's GOP and are naturally at odds with the man at the top, those two video segments are both presenting deceitful arguments. Corker and Costello both bring up the $25B border security bill proposed by Dems, but reports were initially unsure of the actual figure, sources like Washington Post at the time had said it could be $20B. So now you say, "But $20B or $25B is still a lot more than the $5B he's asking for today." No, it's not because that $20-25B bill wasn't going exclusively to the wall and it was spread out over a 10 year period, a point that rarely ever comes up (case in point, your two videos) because it hurts their position. Schumer and the Dems received a lot of blowback for it and just like Trump reversed his decision to sign this month's spending bill, Schumer rescinded his offer, too.
Costello brings up "a couple of talk show hosts" as the reason for Trump's reversal, which isn't technically a lie, but is far from the whole truth, too. Spend a few minutes browsing social media reactions and you'll see the anger this raised among his base or jump head-first into The_Donald to get it from his most hardcore supporters. He was going to take a serious hit if he signed that bill.
His hardcore supporters would have fallen in line after a day or two. They twist reality to excuse everything else he does. What you see as people upset they no longer have a place in Trump’s GOP, I see as people upset Trump’s GOP exists. Luckily the rest of the GOP appears to be starting to agree.