"The real problem with land [and property taxes] is the fact it tends to flow into the cost of rent and homeownership."

That's simply not true in the case of rent payments to landlords if we're talking about a land-value tax (i.e. taxing a proportion of the unimproved value of land). Land-supply is effectively fixed in the short-run (and pretty damn inelastic in the long-run thanks to over-regulation, zoning etc.). Therefore, tenant rents are determined solely by what the demand-side of the market will bear, meaning landlords can't pass LVT on (so they bear the economic incidence of the tax). Just to clarify terminology, by land-value tax (LVT), I mean taxing the proportion of unimproved land value that represents capitalisation of future economic rents (unearned profits) from land.

It's plausible a LVT would actually decrease rents paid by tenants, as it would increase the elasticity of dwelling supply in the long-run (land-owners would have stronger incentives to use land more productively). A properly designed LVT only taxes economic-rents (unearned profits) that currently accrue to private land-owners: if the local council builds a park near some land, the increased amenity for the surrounding area translates to increased economic rents, as the increased amenity is captured in the form of increased unimproved land values.

As for home-owners (existing land-owners), theoretically if people are unable to move houses then LVT would increase the cost of home-ownership: directly from increased tax and indirectly from decreased value of their land holding (as future economic rents from land is capitalised into land values). However, this is an unlikely scenario. More likely, the revenue raised from LVT would hopefully be used to cover reductions in property transfer taxes (i.e. ad valorem/stamp-duty taxes), which have absurd efficiency costs. This would reduce the cost of home-ownership overall.

People would pay less transfer tax over their lifetime, and property and land prices would fall due to increased allocative efficiency: reduced transaction taxes mean more efficient allocation of existing housing and land stock, as the tax disincentive to reallocate is reduced or eliminated. Also, under an LVT existing land would also be used more efficiently (e.g. medium-high density inner-city dwellings) as there would be higher opportunity costs of using high-value land unproductively (e.g. building a 1 bedroom shack on a big plot of land in the CBD).

In my opinion, the real problem is that these dynamics aren't immediately evident, are difficult to explain, and it's very easy to (mistakenly) conflate legal incidence with economic incidence of taxation. It's interesting you bring up gasoline taxes as this is the classic example: the Carter administration attempted to increase gas tax and hand the revenue back out through tax-cuts. It didn't happen because many thought it was simply taking with one hand and giving with the other.

In actual fact, given the aim was to reduce US oil consumption, it was pretty sound policy. Even though people would have ended up with the same amount of money (neutral income effect), gasoline consumption would have been reduced due to substitution effects: gas would have become relatively more expensive in comparison to every other consumption possibility, meaning people would substitute away.