It depends on a lot of factors you haven't revealed in your question. Among them:
- How does one define "practical?" For some that would be "nothing less than the fastest possible speed available by all current options." For others it may mean "anything that can do it unattended while I'm asleep for the night."
- What size are your 5,000 photos? Are they low resolution, highly compressed jpegs that are about 100 KB each? Or are they high resolution raw files that are 50+ MB each? Or somewhere in between?
- Likewise, what compression codec, if any, is being used for your 4K video? This can have a significant impact on file size per minute of video. How long is the video footage in question?
- Does requiring use of an AC power adapter to power your camera for the amount of time it would take to transfer a large amount of data affect your idea of suitability? How about tying up the computer you have attached to the camera's WiFi for several hours?
In general, it would be highly impractical to transfer 5,000 images per day or any amount of 4K video over a camera's built-in WiFi capability.¹ There are some cameras with built-in WiFi that only allow still images to be transferred via WiFi and do not allow any high resolution video to be transferred via WiFi.
I've never seen a camera that exceeded the 802.11n standard, which is a theoretical maximum data rate of 450 Mbps or 56 MB/sec. In practice most such connections are slower due to overhead, encryption, error correction that requires two-way communication, etc. Even at ≈300 Mbps, it would take around 1.5 seconds per ≈50MB unencrypted image. That translates to a little over two hours for 5,000 images. If your camera's battery lasts that long.
The impact of encryption can typically be anywhere from 5% to 50% in actual transfer speed, depending on the capabilities of the processors used in each device to encode/decode the encryption. So now we are talking up to a little over four hours for 250 GB.
External USB card readers are cheap and practical. Another alternative to consider is a wireless hard drive with a built in SD card slot. They come in both spinning disk and SSD flavors. This would allow you to access the images as needed using a device with no USB or SD connections. Such a device would presumably also have a limited amount of internal storage space. This would be a problem if you are generating 250-300GB of content per day!
Is it easy to find USB-C cables for some known professional cameras?
That can only be answered on a camera-by-camera basis.
"Flagship" cameras such as the Canon EOS 1D X Mark II (as well as most of the 1-series digital bodies that preceded it) and the Nikon D5 (as well as most of its predecessors) have ethernet ports built in. Some are 10/100 speed, the newest ones are gigabit ports.
Many pro-level bodies have standard female Mini B USB 2.0 connectors (not to be confused with Micro B USB 2.0 revised connectors) or female Micro B USB 3.0 connectors. A few have proprietary connectors.
In the past I've not noticed any references to a "pro-grade" camera that has a Type C USB 3.1/3.2 connector. Many phones, tablets, netbooks, etc. have them. Some lower end consumer grade endoscopes and thermal imaging devices meant to be used with such devices (phones, tablets, etc.) have Type C connectors.
However, a simple google search indicates that (back in 2016) before they were released the Olympus E-M1 Mk II (first shipped in December 2016) and the Panasonic Lumix GH-5 (first shipped March 2017 ) were rumored to have such connectors. The rumors proved to be true in both cases. Both are Micro Four-Thirds format cameras. One can only assume that eventually more pro-grade (e.g. FF and APS-C) cameras will also sport them. It took the camera industry several years after USB 3.0 was common on computers for the connector and the increased speeds it brought to be included on mainstream cameras. Development times of pro level cameras tend to be longer than other digital devices.
¹ On the other hand, although it would be less impractical to transfer 5,000 images per day using either an USB connection or a card reader, it would still be a pain. Handling 5,000 images per day is a chore no matter how you slice it.