... to avoid the LIKE operator where INSTR is sufficient

Some databases do even better: in postgres there is the option to index on regular expressions (or any string) via the pg_trgm contrib extension.

Its disadvantage is huge index size but the gain can be enormous, as indeed in this, the OP's example:

-- table t is 112 MB, 1 million rows.  The index is 190 MB (!)
-- all queries measured on second run (to avoid uninteresting cold cac
+he slowness)

-- no index:
SELECT * FROM t WHERE name like '%Franklin%' and name like '%Linsey%';
Time: 118.004 ms

SELECT * FROM t WHERE position('Franklin' in name)>0 AND position('Lin
+sey' in name)>0;
Time: 444.716 ms

create index t_trgm_re_idx on t using gin (name gin_trgm_ops);

-- with index:
SELECT * FROM t WHERE name like '%Franklin%'  and name like '%Linsey%'
+ ;
Time: 1.582 ms

-- postgres specific regex-search syntax with '~' 
SELECT * FROM t WHERE name ~ 'Franklin' and name ~ 'Linsey';
Time: 1.895 ms
[download]

The latter syntax has the added advantage that one can use regex like 'Frankl[iy]n' (it will use the trgm index as well).