The .local name is intended for multicast name resolution. RFC6762 says implementations may coalesce results with unicast DNS lookups, but it's not required. The only thing required is that anything ending in .local must he looked up via mDNS. So including .local in the search list doesn't seem likely to make your query reach DNS on an mDNS capable stub resolver.
Meanwhile, RFC4795 says that LLMNR senders should not send queries for single-label domains to DNS and that no search list should be applied to such. So querying a single-label domain does not seem likely to reach DNS on an LLMNR capable stub resolver.
My advice to one who wants to use DNS to resolve hostnames on the local network: Avoid using a domain reserved for mDNS domain and use another one.
The thing is that a simple name doesn't work either, so 'server1' doesn't work, 'server1.local' doesn't work. 'server1.example.com' works, and I think 'server1.lan' also works, but I'm not sure about that last one.
Meanwhile, RFC4795 says that LLMNR senders should not send queries for single-label domains to DNS and that no search list should be applied to such. So querying a single-label domain does not seem likely to reach DNS on an LLMNR capable stub resolver.
My advice to one who wants to use DNS to resolve hostnames on the local network: Avoid using a domain reserved for mDNS domain and use another one.