If particles get mass from the Higgs field, why do we not see Brownian motion?

The cartoon of particles getting their mass by repeatedly bumping into Higgs field is truly misleading--to the extent that it has led the OP to ask this question (which to most high energy physicists sounds "silly").

Here, I can offer another slightly better cartoon that addresses the conceptual issue:

Imagine the Higgs field as a large collection of closely-spaced stumps jutting out of the surface of a vast body of water. Water waves (representing the quantum massless particles in this analogy) attempting to traverse the area with stumps will be multiply scattered, causing them to be "slowed down" which looks like these waves corresponding to particles have acquired a mass. Since we have have replaced the massless particles with waves, this cartoon offers a "smoother" picture which makes it easier to discard the notion that any Brownian motion would occur.

Please keep in mind that this is just another cartoon: every analogy that highlights a particular feature (in this case, the lack of any Brownian motion in the Higgs mechanism) always falls short in something else (the loss of wave packet localization due to the numerous haphazard scatterings, and the apparent loss of momentum conservation as the waves are scattered).

In reality, the waves undergo only forward-coherent scattering with the Higgs field. The "coherent" aspect of scattering holds the wave packet together. The "forward" scattering aspect means that the wave number (i.e. the momentum) remains unchanged. It works just like how glass changes the dispersion relation of light waves due to forward coherent scattering with the glass molecules.

None of this is particularly tied to the fact that it's a left/right chirality flipping interaction.

That's an interesting question. As @JgL says, particles that interact with the Higgs field (e.g. electrons) experience a drag, which makes them massive. You wonder whether these interactions could impart momentum to the electron.

Let's extend the analogy to a ball being thrown through water. If the water is sloshing about, as well as providing drag, the waves could impart net momenta on the ball, making it deviate from a straight line. This is only an analogy; there are aspects of drag (e.g. energy loss as ball slows down) that is quite unlike the Higgs field.

However, the part of the Higgs field that provides drag, denoted by $v$, resulting in masses, is homogeneous (i.e. unchanging) in space and time: $$ \text{Higgs field}(x) = h(x) + v $$ That means there is no sloshing or waves that impart momentum on the electron. The homogeneous part of the Higgs field is like perfectly still water that provides drag and nothing else. The other part of the Higgs field, $h(x$), results in physical Higgs bosons.

The physical Higgs bosons, $h(x)$, could indeed impart momenta; however, this would be nothing but collisions (elastic scattering) between Higgs bosons and electrons. As there is no big source of background Higgs bosons, the probability of any interactions is negligible, and it doesn't cause the electron to deviate from a straight line.

Finally, concerning the effects virtual Higgs bosons (as opposed to physical Higgs bosons considered above) on the motion of an electron. Virtual Higgs bosons cannot be present in the initial or final states. We begin with an electron and end with a electron. By conservation of momentum, it cannot have changed momentum, i.e. it does not deviate from a straight line due to virtual Higgs bosons.

The Higgs field is a field that permeates all of space (very much like the electromagnetic field), Higgs bosons are 'excitations' of this field. In some contexts, you can roughly think of these excitations as ripples or waves in the field, though this is a bit of a sloppy analogy. I would say these 'excitations' have a much richer structure than just being simple waves, which allows them to also behave as what we normally think of as particles.

When the Higgs field interacts with other elementary fields, the excitations of these other elementary fields (which you can think of as other elementary particles) experience what you can think of as 'drag', because due to their interaction with the Higgs field they have to 'plow through' this Higgs field while they are moving through space. This is the typical way the Higgs field gives what we normally think of as 'mass' to e.g. electrons (the excitations fo the electron field). If this interaction would not be there, the excitations of the electron field would be free to move through space at the speed of light.

(Sometimes people prefer to explain Quantum Field Theory in terms of particles to avoid having to mention 'excitations' or 'fields'. When they do this they often talk about 'virtual particles' where they are actually talking about interactions between fields. This, unfortunately is a problem that arises often when you try to explain things by analogy. Your explanation will only seem to make sense as far as your analogy holds. )

Edit: If you know some electromagnetism you can maybe try to play around with two classical fields $\phi$ and $\chi$ who's interaction is described by the Lagrangian $$ \mathcal{L} = \partial_\mu \phi \partial^\mu \phi + \partial_\mu \chi \partial^\mu \chi + \phi^2 \chi^2 $$ If the coupling term $\phi^2 \chi^2$ wasn't there, waves in the two fields could freely propagate at the speed of light. The behaviour of both fields would be described by a differential equation of the form $$ \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = \nabla^2 \phi $$ The coupling $\phi^2 \chi^2$ causes waves in the $\phi$ field to interact with the $\chi$ field. Waves in the $\phi$ field will as a result experience some form of 'drag' (not in terms of friction, but as a consequence of the way the fields interact with each other). When we turn this coupling on, the above differential equation that describes the behaviour of the $\phi$ field will be changed, an be of the form $$ \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = \nabla^2 \phi - \chi^2 \phi $$ The amplitude of the $\chi$ field at a point $x_i$ now behaves as an effective mass that is experienced by the waves in the $\phi$ field at that point. This is analogous to the way the Yukawa coupling of the Higgs field gives an effective mass to many other fundamental fields.