#Short answer Instead of SEARCH use REGEXMATCH and JOIN.

#Explanation SEARCH is for searching partial strings into other strings and if it will be used with arrays, both parameters should be arrays of the same size.

JOIN could be used to create a regular expression to be used by REGEXMATCH which in turn will return a array of TRUE/FALSE if the first parameter is an array and it is nested inside a ARRAYFORMULA.

Considering the OP examples, the formula to be used is

=ArrayFormula(REGEXMATCH(E1:E4, JOIN("|",{"Tim","Frank"})))

Short answer

Instead of SEARCH use REGEXMATCH and JOIN.

Explanation

SEARCH is for searching partial strings into other strings and if it will be used with arrays, both parameters should be arrays of the same size.

JOIN could be used to create a regular expression to be used by REGEXMATCH which in turn will return a array of TRUE/FALSE if the first parameter is an array and it is nested inside a ARRAYFORMULA.

Considering the OP examples, the formula to be used is

=ArrayFormula(REGEXMATCH(E1:E4, JOIN("|",{"Tim","Frank"})))