Are logins with button pushes more secure than normal form logins?

You almost answered your question yourself, I might just try to give some grounds for why it is actually less secure than regular log in forms.

Long story short, is this login method actually more secure? For me, it doesn't seem so, and it's so annoying given that it doesn't work with my password manager.

There is a rule, formulated in security.stackexchange:

Security at the expense of usability comes at the expense of security.

See here.

Called: AviD's Rule of Usability, or how I call it - security through unusability. It basically means that as soon as you decrease the usability of the security measure, security itself will decrease. We can see many such "inventions" in software world:

• disallowed "paste" in the password/username field - I have made fun of it in my tweet: https://twitter.com/m1ndas/status/789984433973628928 You can also read Troy Hunts blog post on that: https://www.troyhunt.com/the-cobra-effect-that-is-disabling/
• crazy limitations on password length;
• strict requirements on special symbols and simultaneous neglect on password length (incentive not to use long memorable passwords).