The trick is that we are using an older school mentality. This is not the 3.5 mentality of "its not a specific option or rule you can't do it". Its "do what you want, and you and the dm work it out together."
So for example, grapple only consumes 1 attack. And while people snide it for being weak, its relatively simple and easy to do, and if you have two attacks only requires a bit of your offense.
The rules give some basic notes for bull rushes, disarms, and trips...no reason you can't attempt those.
In my last game, the barbarian fought a lich with a staff of power at one point. The barbarian rushes in, dashes past the guards, grabs the staff out of the lich's hands (a "disarm" attempt), and then with his second attack we used the "use an object rules" to have him break the staff in front of all of them. He happened to randomly have a ring of force resistance, so only took half from the massive explosion. The lich and all of his cronies were dead, and the barb stood strong.
5e allows for all of that, even encourages it, and gives basic mechanics (such as opposed athletics checks) to administer them. So don't think of it of "there are no options", think of it as "the rules don't say I can't....so!" You and the DM work out it out, and with a little bit of creativity and flexibility, you can absolutely make a fighter doing all sorts of tricks.