Yes, keyservers still exist:
Both the SKS Keyserver Pool (stats) and PGP Global Directory are still online. (Note that the PGP Global Directory is not part of the pool.)
Some pool keyservers, such as Ubuntu keyserver, have replaced SKS with more modern and reliable software such as Hockeypuck. They do however still synchronize with the pool.
New standalone servers are showing up, such as keys.openpgp.org. They are not part of the SKS pool and do not synchronize with each other (at least for now).
People usually use the SKS pool, since it consists of many servers which synchronize their databased continuously. Meanwhile, Global Directory is a single, commercially operated server which may go down at any time; the same goes for the new non-SKS keyservers.
However, SKS has the problem of accepting anything and storing it forever (much like a blockchain). This has caused problems for a long time, but started getting massively abused in 2018–2019. The new keyservers don't have synchronization partly because they want to figure out how to combine opposing goals.
pgp.mit.edu has finally upgraded to SKS and is now part of the pool. There also exist a bunch of other keyservers not part of the SKS pool (listed in the same status page). The default keyserver for GnuPG,
keys.gnupg.net, is now an alias to the SKS pool as well.
Another widely known server,
subkeys.pgp.net, is not part of the SKS pool since (AFAIK) it still runs a very old version of PKS instead. (It also seems to be down, although the website is up.)
If your email address is at a domain name you manage (i.e. can have arbitrary DNS records created), it is also possible to publish your PGP key using DNS. The easiest method for that is PKA, which only requires the ability to create TXT records; see the article on publishing PGP Keys in the DNS.
PKA, as well as two other methods (CERT and IPGP CERT), are described in this guide in much more detail.
One downside of all three methods is that GnuPG must be manually configured to use them, and PGP.com doesn't even support using DNS. Meanwhile, practically all versions of PGP and GnuPG can use keyservers.
Note: GnuPG 2.1.3 has completely changed the PKA format (into a mix of CERT and old PKA).
Given that GnuPG did this in a minor release without any worry about backwards compatibility with the old format (in fact, the old format used to outright crash 2.1.x for a while afterwards), I'm no longer comfortable suggesting pubkey publication in DNS. It's a waste of time. Use keyservers.