I read that one of the benefits of VXLAN over VLAN is that it can spawn across WAN and multiple layer 3 networks by creating overlay layer 2 networks.
However AFAIN VLAN can also do the same using MPLS VPN, L2TPv3 or GRE tunnels, or by trunking and using VRF if you own all the hops in between.
What is the real benefit of VXLAN over VLAN with regard to cross layer 3 communication?
Yes, from the packet switching point-of-view, VXLAN is just a matter of sticking some encapsulation on top of an L2 frame: something that other protocols do as well.
The real difference it makes is at the control and management layer.
VXLAN evolved as a Data Center technology, so the ability to span a WAN is just an additional advantage, not the thing that drives the technology.
Consider a cloud service provider, with a data center that can contain thousands upon thousands of virtual machines. These VMs can belong to different enterprises (the cloud provider's customers), and all doing different things, from running e-commerce applications, online shopping, ML/AI applications (like suggesting you what to buy for your wife for her birthday :-), managing calendars and meetings and so on.
In an environment like this, the 802.1Q VLAN limit of 4096 is laughably inadequate. The data center admins need a way to segment their network in more flexible and fine-grained ways.
Also, unlike say an enterprise's network wiring, which follows a hierarchical model (access -> distribution -> core), the devices in the data center need to be wired up in a more-or-less flat manner.
So basically imagine a huge flat LAN with a very large number of hosts.
Next, you also want to provide redundancy - protection against failure of individual switches and individual links. Spanning tree is of course a non-starter here: we want every link spewing data close to its max capacity. Hence the IP-based fabric, and the good things that IP comes with (like routing protocols, equal-cost multiple path support).
Next, when you get a new customer for your data center, you want to be able to deploy their VMs ASAP (in hours if not minutes), which means you want to add a new switch to the fabric without disturbing the existing switches. So, in a fabric that contains 77 switches, when you add the 78th, you most certainly do not want to spend time provisioning 77 L2TPv3 tunnels :-)
Hence the first line from Wikipedia's VXLAN page: "Virtual Extensible LAN (VXLAN) is a network virtualization technology that attempts to address the scalability problems associated with large cloud computing deployments"
As you've already stated, VXLAN is L2 tunneling over IP. It's a solution to use any L3 network for creating a L2 segment.
While this is also possible with other protcols, VXLAN doesn't require additional infrastructure or special transport (given IP is available) and it can also use a single tunnel for up to 16 million subtunnels - with a large infrastructure, the latter can make a great difference.
