No; Charger removes the prone option.
It's right there in the text: After fulfilling the movement requirement, if you choose to shove and succeed, you push the target 10 feet away. That's the full effect of the enhanced shove, replacing the normal effects (including the option to knock prone), not in addition to them.
It wouldn't make sense to interpret the Charger "10 feet in a straight line" benefit as in addition to the normal results of a shove (i.e. a choice between knocking the target prone and pushing them back); that interpretation would imply that you could push a creature 5 feet and 10 feet at the same time by choosing the "push" effect. It seems clear that the intent of the feat was to allow you to push a creature 10 feet instead of the normal effect, and if that's the case, then it makes no sense to interpret it as letting you knock the target prone and push them simultaneously.
As a DM, I'd certainly allow a player to choose to knock the target prone with the charging attack, but they wouldn't gain any benefit beyond the normal shove effect of knocking prone. They'd just be choosing between knocking the target prone and pushing them back 10 feet, not both.
Hammering Horns is its own specific action with its own rules
As you've quoted, the Hammering Horns trait of the minotaur player race is specific in how it works and what it does. It unequivocally states that you start with 'unlocking' the ability by taking the Attack action (my emphasis):
Immediately after you hit a creature with a melee attack as part of the Attack action on your turn, you can use a bonus action to attempt to shove that target with your horns.
After you've unlocked the ability, you can now attempt the Hammering Horns trait, which states:
Unless it succeeds on a Strength saving throw against a DC equal to 8 + your proficiency bonus + your Strength modifier, you push it up to 10 feet away from you.
Yes, it's called a shove, but it is not the standard PHB shove. It is Hammering Horns, which gives you a bonus action to push someone up to 10 feet away from you.
So is this the Shove listed in the PHB or not?
Well, it's not. It's close, but it's inherently a different action because the results are different.
A standard shove is only a 5-foot shove. Hammering Horns lets you push someone up to 10 feet. As I've noted above, it also doesn't state you can knock prone.
But the real difference is in that pushing distance. If this was an error, it would be much more likely that the push distance would be the same 5 feet.
In this, the ability gives you greater range for shoving, but at the expense of no option for prone.
But that's not all. It's also not an opposed contest, but a Strength save.
In all, it's a different mechanic than the PHB attack shove.
But that's silly! Why wouldn't I just use one of my attacks in the Attack action to shove?
Well, you can! But you're giving up a weapon attack to do it. Hammering Horns allows you take all of your normal weapon attacks you have in the Attack action and then grants you a bonus action that you normally don't have. And that bonus action only can shove 10 feet away as the ability states.
Best Answer
The feat does what it says it does
The Telekinetic feat enables you to attempt to shove a creature, specifically to move it 5 feet closer or further away. The feat does not enable you to shove to knock the target creature prone, as opposed to shoving as a bonus action using the Shield Master feat (which does not specify whether you have to push or knock prone).