Approving child's app download without an apple device

From: Approve what kids buy with Ask to Buy - Apple Support

See the section below for how to get approval in person.

How children or teens can make a request with Ask to Buy

1. To buy or download an item, tap the price or Get button. Or if you're using an Apple Watch, double-click the side button, then enter your passcode.

2. If asked, enter your Apple ID and password.

3. Tap Ask. If your parent or guardian is nearby, you can tap "approve it in person" and they can approve the purchase directly from your device.

4. After your parent or guardian buys the item, it downloads to your device automatically.

You tapped "ask" and it doesn't show? Have you scrolled down?

Refer to this page for Apple Support features ➔ Contact - Official Apple Support

Select from the presented options until you find a solution for your issue, or see if there is a chat or phone call contact method offered lower on the page (you sometimes have to narrow down the options multiple times before this is shown). If you do not see your issue, keep experimenting with any series of selections until you reach one that offers a chat session or a telephone call and get the representative to redirect you.

This support article has various country telephone numbers for contacting Apple for support and service ➔ Contact Apple for support and service - Apple Support

How children or teens can make a request with Ask to Buy

1. To
2. buy or download an item, tap the price or Get button. Or if you're
3. using an Apple Watch, double-click the side button, then enter your
4. passcode.
5. If asked, enter your Apple ID and password or your device passcode.
6. Tap
7. Ask. If your parent or guardian is nearby, you can tap "approve it in person" and they can approve the purchase directly from your device.
8. 
9. After your parent or guardian buys the item, it downloads to your device automatically.2

Sounds to me like you need to talk to a different Apple person. Having a child hand you their device for you to then enter your account information (or whatever) for permission doesn't exactly sound like a technology incompatibility issue. Does your child have a child's level account or an adult/teenager account? There could conceivably some nuance if your account was simply set up using the Apple TV. I don't have an Apple TV or a child so I'm kind of left guessing here. See if you can get somebody at Apple who is more familiar with this. Since you set up the account and Family Sharing it sounds to me like there shouldn't be any limitations on your account.

What did Apple say when you called them using the information in the link I provided?