iOS 15 introduced the Find My feature that works even when the device is turned off. The applicable models are: the iPhone 11 through iPhone 15.

^{Source: The Verge Guide to the iPhone}

Find My introduces new capabilities to help locate a device that has been turned off or erased, as well as live-streaming locations for family and friends who choose to share their location. Separation Alerts notify a user if they leave an AirTag, Apple device, or Find My network accessory behind in an unfamiliar location…

^{Source: Apple iOS 15 Press Release}

You can't enable Location Services remotely. With LS disabled, you can't track the phone continuously, but if you set Lost Mode on icloud.com, you may still get intermittent updates from other Apple devices that happen to be nearby:

The Find My network is an encrypted, anonymous network of hundreds of millions of Apple devices that can help find your stuff, even when it's offline. Nearby devices securely send the location of your missing device to iCloud, so that you can find it in Find My.

If Find My shows "No location found" or "Offline" (support.apple.com)

Although I can't find any official documentation, according to at least one user report LS will be enabled temporarily to send a "last location" fix if the phone is in Lost Mode and it connects to a Wi-Fi network. To be certain, you'd have to test this yourself.

Does send last location work with location services off? (apple.stackexchange.com)