What project estimation/management process would you suggest as an alternative?

Just rank by date needed order on a kanban board and work your way through everything in order. If it's constant fight to meet deadlines it will be clear enough that things are backed up.

Its not so much as estimation as a whole work style, but

Make it work -> Make it right -> Make it fast

is arguably the best way to go about things, then structure your work around that.

Ive done this with greenfield projects when I used to work for Amazon (while still ironically operating under scrum), and was able to get SD2-SD3 promo within 2 years (entering an an SD2).

In terms of planning work, you basically allocate people as necessary. The first part deals with a lot of unknowns, so estimation is pointless - basically everyone is on board in terms of getting software up and running and talking to other software.

Once you have that, making it right is a lot easier to estimate because you can do a lot more fine grained planning (like for example, a certain team member that worked on a feature can add all the correctness and unit testing way faster than someone who has not)

Then making it fast is basically just optimizations, which can be done by a subset of team members while the rest work on adding features (and adding features needs to be done in the same way - make the feature just work, make it correct for all use cases, and then make it optimal)