The Senate is an interesting political body, because the minority party has (had?) tools available to check the majority. In the past, there were two different types of filibuster: filibuster on appointments, and filibuster on legislation. These filibusters allowed the minority party to scuttle the majority party's agenda. And overcoming a filibuster required 60 votes, not 51.
Harry Reid blew up the filibuster on appointments during the Obama years, when Democrats grew frustrated with Republican attempts to block all Obama's appointments. He left filibuster on Supreme Court appointments in place, but eliminated the maneuver for all other positions. This gambit was dubbed "The Nuclear Option," as it basically triggered a power struggle battle royale that's continued to this day.
Once Republicans seized control during Trump's years, Mitch McConnell responded to Reid's move with one of his own: he blew up the filibuster on Supreme Court appointments. This is why Democrats were powerless to stop Gorsuch, Kavanaugh, and Barrett. (It's also why Republicans will be powerless to stop Stephen Breyer's replacement from going through, so stand by for those fireworks!)
That leaves the filibuster on legislation as the only tool available to the minority. Because the legislative filibuster stands, Democrats will not be able to pass any legislation through the Senate without Republican help. They only have 50 votes, and 60 are needed to vote down the filibuster. Harry Reid had more than 50 votes, so he was able to simply change the rules and eliminate the filibuster on appointments. Mitch McConnell had more than 50 votes, so he was able to change the rules and eliminate filibuster on Supreme Court appointments. But Chuck Schumer only has 50 votes. Remember, the VP only gets to vote in the case of a tie. And even though there is a 50/50 split in the Senate right now, there are a few Democratic senators who simply will not vote the party line for the sake of "party agenda." Hence, Chuck Schumer cannot simply change the rules and eliminate the legislative filibuster.
Senator Joe Manchin of West Virginia has become somewhat famous for frustrating his fellow Democrats with his truly centrist voting record. He is on record rejecting the idea of changing the Senate rules to eliminate the legislative filibuster, stating "We've harmed the Senate enough."
Considering Manchin's stance, Democrats are actually 49/51 at a maximum when it comes to overcoming minority opposition and ramming through sweeping left-wing legislation. That already takes Harris' vote as VP off the table, since there is no tie.
Also, there's a new Manchin in town! Kyrsten Sinema of Arizona has also indicated she'd vote against changing the rules on overcoming legislative filibuster. She's allied with Manchin on this stance.
Considering where Sinema stands, Democrats are actually 48/52 at a maximum when it comes to overcoming minority opposition and ramming through sweeping left-wing legislation.
Democrats cannot overcome the legislative filibuster with the party's current Senate personnel, and Kamala Harris' vote is moot on the issue because there is no tie. The only way they could pass legislation would be to actually negotiate well enough, in good faith, to peel away 10 Republicans to vote with them.
So hopefully that helps you understand why 50/50 control in the US Senate necessitates a power sharing agreement right now. Senators are individuals, and some of them are (thankfully) more loyal to their constituents than to their parties, meaning 50/50 membership doesn't necessarily mean 50/50 votes. Neither party truly has the upper hand right now. Civility and negotiation must be the name of the game to get anything done, and the Schumer/McConnell talks are the very beginning of restoring the requisite level of decorum to our government...hopefully.
Update: In light of recent political happenings, a person might have questions as to how a $1.9T package was just passed through the Senate on a party-line vote, considering what I've written above. Why were 51 votes all that were necessary? What happened to these claims that Democrats would have to negotiate with Republicans? The answer is actually very simple: Democrats used the process by which Congress sorts out the budget in order to pass this through. That process is called "budget reconciliation," and the legislative filibuster does not apply to it because there is a baked-in limit on debate during budget reconciliation.