SQ for travel on UA. Although J availability can be hit or miss.
UA for no-YQ with good availability on a variety of airlines especially close-in.
If you have a Chase United co-branded credit card UA is probably your best bet as you have better chances to find availability. The best part is that transfers are instant so you can wait to transfer until you're ready to book.
After accounting for YQ I am not sure SQ is such a great option (except for UA which has very little saver availability in J) I am not sure the math makes sense vs buying a revenue ticket on TK (and sometimes if you're willing to start travel outside of NYC (such as YTO, WAS or BOS) you could get an excellent deal on a DL or UA J fare to TLV that connects in NYC (giving you a non stop NYC-TLV in a widebody J, rather than switching in Europe to something else).