I would be very cautious when introducing any algorithms other than the standard methods. The standard algorithms are standard for a reason: they're easy to set up and hard to mess up.
The lattice method, for instance, is hard to set up (it takes a lot of time and effort to draw the entire grid with diagonals) and easy to mess up (if I had a dollar for every time I saw a student draw a sloppy grid with misaligned diagonals and screw up the calculation as a result, I'd be rich). Note that while I'll be focusing on the lattice method in this answer, the same argument applies to the "straight lines" method, which seems even harder to set up and even easier to mess up.
Kids will often latch onto whatever method they "like" best, regardless of its practicality -- and their incentives are often misaligned. For instance, I've tutored students who straight-up told me they preferred the lattice method because they liked being able to take a break from math to draw the grid (and believe me, they took their sweet time drawing the grid and making it "perfect"). Of course, it took these students forever to complete their problems because they were working with incredibly low efficiency, and that frustrated them, but they were resistant to switching to the more efficient standard method because they had completely forgotten it (as a result of using the lattice method for so long) and relearning it would require some additional up-front time and effort on top of what already felt like an overwhelming workload.
I'm not against alternatives, but as a teacher you have to run through a simulation in your mind: "what will happen to students who latch onto this method indefinitely and resist using other methods?" If the alternative method is just as efficient and just as general, then sure, introduce it. But if not, then I wouldn't introduce it, because students who latch onto it and resist letting go are going to be in for a world of hurt. (Even if you introduce an alternative method as a fun, temporary vacation away from standard techniques, some students will try to stay on that vacation forever.)