It's always the type you don't have, or the experience you don't have, or the time you don't have, that doesn't get you the job that you want.
Go get a type rating, and you'll lack the experience and time in type. Go waste your money on a little time in type, and you'll be competing with people having considerably more time in type...your type rating and the time that goes with it will be but a drop in the bucket and hardly competitive.
Go get that 737 type. You'll be competing with a massive pool of pilots with 737 types, many with substantial experience. The only problem is that only a handfull of places expect you to come on board with a type rating...especially B737 operators. The rest pay you to go get your type, and your arriving having a type rating won't help your chances at all.