I'm wondering what serious photographers are using to handle this type of scenario these days?

Canon's 1-Series, 5-Series, 6-series, and 7-series have optional hardware or built-in Wi-Fi to do this. The Wireless File Transmitters for models in this series are rather pricey. They've been available for the 1-Series and 5-Series cameras since 2004 and for the 7-series cameras since its introduction in 2009.

Some of the newest models, such as the 6D, 6D Mark II, and 5D Mark IV, have Wi-Fi and even Bluetooth or NFC built in.

The 7D Mark II, the 5Ds and 5Ds R don't have built-in Wi-Fi, but are compatible with the very affordable W-E1 Wi-Fi Adapter that fits in the SD card slot on these three camera models, each of which also includes a CF card slot as well.

Beginning with the 70D, the x0D Series has had built-in Wi-Fi. Prior to the 70D, the x0D Series had Wireless File Transmitters available beginning in 2004 with the 20D. Though the more recent WFT units for the x0D series are not as pricey as the WFTs for the xD Series, they were still expensive compared to an Eye-Fi card.

These are the types of cameras 'serious' photographers tend to use if they need to transfer files wirelessly as they take them.

Canon's lower end cameras, such as the Rebel line, did not offer such functionality until the Rebel T6i/750D was introduced with built-in Wi-Fi, NFC, and low power Bluetooth in 2015.

If not, should I be looking to eBay to buy a used Eye-Fi? Are those even mildly supported any longer?

Your Rebel T5i/700D is compatible with and supports Eye-Fi cards.

From the comments:

Point taken, though Wi-fi enabled cameras are still relatively new. Many pro photographers probably don't have them yet. I wonder what those who want Wi-fi connectivity are doing?

The 1-series and 5-Series has had compatible Wireless File Transmitters available since the WFT-1 was introduced in 2004.

The 6-Series has had built-in Wi-Fi since its introduction in 2012.

The 7-series has had WFT units available since introduced in 2009.

The 20D through 50D have had WFT units available since 2004.
The 60D (2010) had no available WFT unit but was the first x0D model compatible with Eye-Fi.
The 70D (2012) and 80D (2016) have built-in Wi-Fi.

Additionally, many pro and enthusiast Canon models have dual card slots with at least one SD/SDHC/SDXC slot:
The 1-Series from 2004 until 2012 (the 1D X (2012) has dual CF slots and the 1D X Mark II (2016) has 1 CFast slot and one CF slot)
The 5-Series since 2012 (the 5D Mark III is compatible with and has many features available via Eye-Fi)
The 7-Series since 2014