If a line is being used the caller receives a busy signal, it doesn't go to voicemail

Call the phone company to fix the problem.

Without call waiting or a second line, callers will get a busy signal if you're already on a call. This happens at the switchboard; it doesn't matter what type of answering machine you have because the second call doesn't get that far.
You can record your own message on the 1739 but, as mentioned above, if you're already on a call then the second call never gets that far.

Contact the phone company. They do not have the lines rolling over or "Hunting". The phone system does not have this kind of control.

Sounds like your service provider(Verizon) has your phone line setup to forward to voicemail on no answer after 2 rings. Call your service provider's customer service number and request that the number of rings for forward no answer be increased to at least 5, then your answer machine can receive and answer your calls after 4 rings but if your line is busy callers can leave a message with your Verizon voicemail service. If you don't want Verizon voicemail to answer any of your calls, you could have them disable the voicemail service. When you call the service provider have a recent bill ready because they may request your account number before they will make any changes.

When you are on the phone the caller will get a busy signal, unless you have call waiting on the line, then the caller will hear ringback. If you have call waiting you can answer the second call by sending a flash signal to the service provider which puts the first call on hold while you are connected to the second call but your answer machine cannot answer the second call because you have only one line therefore only one speech path. So the answer to your question is no. If you have the voicemail feature from your service provider then the caller could leave a message not on your answer machine but in your service provider mailbox.

Apart from needing more phone lines, the main solution was to use the TVM50 Voicemail instructions and not the guide that came with the KX_T7630 so the earlier Panasonic is a good place to start!

turn off dnd

ANY phone system will have this problem.

If 5 people call in, the first 4 will be answered, but the 5th will get busy, as there are no lines available into your phones voicemail system.

IF, you have voicemail through the phone company, you can get their voicemail feature for extra.

THEN, if 5 people call in, the first 4 go to the phones, and the fifth gets the voicemail greeting.

It would be dependant on the line provider how many calls their voicemail would handle at one time. SO if 500 people call in, 4 get through, and MAYBE 5 go to voicemail, the rest get busy.....

Make sure the handset is set in it's cradle. But it sounds like some other device is on the phone line.