where to find replacement Taylor deadbolt cylinder?

Ok, just to update, I managed to remove the cylinder. It has a Taylor patent number on it, but that's all I can see.

The big stores will carry entire locks but doubt they have parts. You may do best with a locksmith that can get a proper replacement or repair what you have.

Chain, weakest link, etc.

How is your lock attached to the door and door frame?

As Bob asked, it would help to know why you want to replace the cylinder.

What is shown in the ebay image is a double cylinder lock - requires a key to unlock from the inside. The only similar double cylinder lock I have seen, the inside cylinder can't be replaced. If the cylinder is not damaged it can be rekeyed. If you don't have a key a locksmith can pick or shim or use another method in rekeying.

If you have just a single cylinder with a thumb turn on the inside lock, the cylinder, that goes through the door, can also be rekeyed. Or it is relatively simple to replace. There are several kinds of tailpiece, the bar out the back of the cylinder through the door. You need to match the tailpiece. The length of the tailpiece needs to be adjusted for the thickness of the door by breaking it at the appropriate point. I have never looked, but cylinders are likely available at hardware or big-box stores. Given your questions I suggest a locksmith. If you removed a single cylinder, the replacement is about the same.

Replacing the whole lock mechanism with something that is not identical is likely to be a pain.

In some locks like this, there is a shutter on the back of the lock. If the cylinder is removed the shutter closes and can't be opened. You may have to work with this feature. Haven't seen it, but my guess is the shutter is held open when the lock is unlocked.

I have double cylinder deadbolts that are locked when I am out of the house.

And just a note on safety...

Locks that need a key to open on the interior side can be a safety issue.

Any lock that is keyed on the interior side should have the key left in the lock in case of an emergency. An emergency is the wrong time to be looking for the key.

Even a hook near the door can be an issue for someone not familiar with the environment or during a fire, etc. In the lock, every time, is the safest place to keep it.

While I agree from a safety point it is also a way to break in. Some years ago my basement door had four panes of glass. Once morning the pane by the lock was broken. Had I left the key in the lock the intruder would have gotten in. That is why I had a double keyed lock. The key is nearby but not reached from outside.

If you have no glass you have no need for double keyed.

They may not be needed, but they do exist.

In urban areas, doors with panes often have bars added but perfectly good double keyed lock sets are not replaced.