I can do this with fastmail, though fastmail is a subscription (like $5/month? IIRC, mine auto renews every 2 years so not sure). I have my primary email setup as <firstname>@<lastname>.org. If you set your dns records correctly with them, that allows you to use without any ahead of time setup <randomtag>@<firstname>.<lastname>.org. Setting a different tag where I have <firstname> is can be done too, but you need to set those up individually.
replying to emails, I can change <randomtag> to whatever I want.
They also offer random domains that you can setup burners under, though that does involve some ahead of time setup.
Fastmail lets you create wildcard identities like this so you can send from any username at any domain you have with them, but if you are sending from a third party app you usually still need to set up the sending identity in the app itself, which is annoying. The email programs I've tried haven't let me type arbitrary addresses into the 'from' line.
Many programs won't even automatically reply from the same alias the message was received at.
replying to emails, I can change <randomtag> to whatever I want.
They also offer random domains that you can setup burners under, though that does involve some ahead of time setup.