The Thunderbolt 3<—>Thunderbolt 2 adapter does not carry just Display Port signals, so it will not work.
An eGPU is an expensive solution to connect a display that old. It would be cheaper to find a decent 4K display, a 1440p or a 1080p display and move on.
Most Thunderbolt 3 docks have moved to a full size Display Port port and so you’ll need an adapter such as this - Cable Matters DisplayPort to Mini DisplayPort Adapter (DP to Mini DP) - 6 Inches
https://www.amazon.com/dp/B014DV2RGG/ref=cm_sw_r_cp_tai_b5HtDbRR97BQR
The cheapest dock with full-size Display Port would be the CalDigit a Thunderbolt 3 Mini Dock
- CalDigit Thunderbolt 3 Mini Dock -Portable, Bus Powered, Non Charging, 40Gbs, Dual 4K Display Adapter, USB 3.0/2.0 and Gigabit Ethernet LAN for Thunderbolt 3 Mac and PC (Dual HDMI 2.0, Black)
https://www.amazon.com/dp/B079H4WTR4/ref=cm_sw_r_cp_tai_zmItDb425XFMT
Again, a high quality 1440p 27” display can be had for $400.00. Or go for something a bit higher grade like the BenQ SW2700PT. Good luck!