Many searchers will be looking for 2+ years for a company and a decent amount of them close their funds unable to find one. Search is also very emotionally taxing. Plus, if you fail to acquire, it becomes (more) difficult for them to find work afterwards / they have to go back to square one.

Why not continue working a job (being possibly paid more) and do a self-funded search (where you also have more flexibility in the acquisition ie one investor can fund it) to eliminate this risk? What are some reasons to pick the traditional search fund route instead of self funded?

Given that you spend two years just looking for a company, for a minority equity stake (if successful), I believe that building a 1M+ ARR service-based company in that timeframe for a much bigger equity stake is also doable without the risk.