Let me try to answer this by problem raised

Password to hard to remember:

You can opt to have a pin number as well. This is only 4 digits and is in the accounts settings of "Signing In".

Watching over Child's account:

You will need to have an email address for the child. Sounds weird but mine have multiple accounts already for when they grow up (Toddler, Child, and last is a family emergency contact where it never gets used or stored except close family). So kids having emails is not unheard of especially due to the email addresses now being like an SSN for the IoT.

Once that's done you setup a Microsoft Account using that email (Or use theirs.. MS's.. if you like) as a child. This add account is in control panel for those not knowing and reading this as a "Help". Go through the steps and make sure when you set it up, watch what's going on and read. Dont just click through. You'll start to notice they are making it so if the child logs in under their name on this computer or any computer your settings you have picked are enforced no matter the system.

Then for the fine tuning goto:

Family Settings on the web.

This will have all the needs you are looking for. It has setting screen times on certain days as well as certain times and also allowing or disallowing websites. So if you find a link to a certain YouTube you can allow it or through his account and email make it so Youtube also helps block by listing the account as a child.

Later as the child grows and you share products like office with them they can use it since they are linked.

Another nice feature is it can track them and useage as well as "Locate" them to see where they are if logged in. (While some claim "OMG MS IS SPYING"... for a parent this can be one heck of a piece of mind feature which also can be turned off. As with most "OMG SPYING" features).

What this also allows is if you travel or they go to a family member/ Friend house. They can still login but it will immediately have the settings you want and you dont have to worry.....much.

This was to answer how to do this in Win10. I use multiple systems and OSes and havent had the "Problems" that most have with Win10. Also to note my one 5 year old uses this and settings but also Linux (Very basic setting with MATE) since he likes the Tux mascot. So I know everything mentioned can be done like you would like since ive done them on his Win10 Rig.

Hope this Helps.

Given the information you've provided, you seem like a fairly competent person and simply need a little direction in how to troubleshoot this issue next, so I'll do just that. Normally, I'd add a comment and ask what I'm about to ask, then give an answer based on your response, but I think we can skip the former.

The only option you have available is to see exactly what is happening to the TCP connection as it fails. There are a couple ways to do this, ranging in complexity and detail. Given that you don't need a whole lot of detail, I'd recommend that you download and install Fiddler, which should provide enough information for you to narrow down what is causing this to happen. However, there are a few things to keep in mind.

1. You said this is a corporate computer, so I'm assuming you're using some version of Windows and have local admin access to it so you can install new software.
2. After installing Fiddler, you'll need to configure it to decrypt HTTPS sessions as YouTube (like most modern websites), defaults to HTTPS. If you can't figure out how to do this, use Google. I'm certain there is, ironically, a YouTube video showing you how to do this. In the off-chance that it is one of this videos that won't load, there's likely written instructions too.
3. You may have to install and configure a browser extension for it to start picking up HTTP(S) data from your browser.
4. Pay attention to the status code of the requests for the videos that do not load. That will help tell you if the conversation is dropping immediately, if it's the remote server dropping the connection, or something else all together.
5. Past Fiddler, the next best utility you could install would be Wireshark, but it is a lot more complex and may be too much for you to navigate without help. Also, running it while on your company's corporate network may be a (serious) network security violation. You have been warned.

Lastly, if this is happening regardless of browser or network, that tells you that it is something specific to either the computer or your account (assuming you're logged into YouTube on both browsers). To rule out various possibilities, you could try creating another user account on the computer and log into it to see if this still happens. You could disable any firewall software/services that are running in the background. There's probably more you could do, but those are the two things that come to mind first. Good luck