What prevents the use of a multi-segment ILS for non-straight approaches?
See here, pages 5-23, 5-24.  Things have advanced well beyond ILS for that kind of approach.

https://whispertrack.com/pdf/faa_handbook.pdf


  REQUIRED NAVIGATION PERFORMANCE
  The operational advantages of RNP include accuracy
  and integrity monitoring, which provide more precision and lower minimums than conventional RNAV.
  RNP DAs can be as low as 250 feet with visibilities as
  low as 3/4 SM. Besides lower minimums, the benefits
  of RNP include improved obstacle clearance limits, as
  well as reduced pilot workload. When RNP-capable
  aircraft fly an accurate, repeatable path, ATC can be
  confident that these aircraft will be at a specific position, thus maximizing safety and increasing capacity
  
  To attain the benefits of RNP approach procedures, a
  key component is curved flight tracks. Constant radius
  turns around a fix are called “radius-to-fix legs,” or RF
  legs. These turns, which are encoded into the navigation database, allow the aircraft to avoid critical areas
  of terrain or conflicting airspace while preserving positional accuracy by maintaining precise, positive course
  guidance along the curved track. The introduction of
  RF legs into the design of terminal RNAV procedures
  results in improved use of airspace and allows procedures to be developed to and from runways that are otherwise limited to traditional linear flight paths or, in
  some cases, not served by an IFR procedure at all.
  Navigation systems with RF capability are a prerequisite to flying a procedure that includes an RF leg. Refer
  to the notes box of the pilot briefing portion of the
  approach chart in figure 5-17.


The Approach in Fig 5-17 has 5 turns left &amp; right!