I recently learned that there are plenty of services on the internet that offer/list free public proxies (freeproxylists.net, hidemyass.com, proxyscrape.com to name a few). Is this safe? Can't an attacker easily connect to multiple of these free proxies at once and essentially build a botnet?
A bot (as in botnet) is something different than a proxy. While attackers can use proxies to hide their own IP address from the target they can not actually control the proxies with small commands to mount large attacks against some target. A proxy just passes through the original traffic, i.e. an attacker would need to have lots of bandwidth by its own to mount an attack against some target with the same bandwidth.
A bot instead provides a way for the attacker to run arbitrary actions. Specifically it is possible to control lots of bots with only little bandwidth, but let these bots then mount bandwidth-intensive attacks against the DDoS target.