This is not a problem with cosmological constant in models of universe. The "cosmological constant problem" is a bad name for a particular (bad) interpretation of certain calculation procedure in QFT.
When we canonically quantize e.g. free EM field, we get a field Hamiltonian that is a sum over Fourier expansion modes (where mode $\mathbf k\lambda$ is a multiindex indexing discrete basis for wave vector and polarization), in which there is a contribution $\frac{1}{2}\hbar \omega$ per mode:
$$ H_{cq} = \sum_{\mathbf k,\lambda} \bigg( \hbar\omega_{\mathbf k} a_{\mathbf k,\lambda}^+ a_{\mathbf k,\lambda} + \frac{1}{2}\hbar \omega_{\mathbf k}\bigg). $$
This zero point term implies all eigenvalues of $H_{cq}$ are infinite, and expectation value of $H_{cq}$ is infinite for any state.
Having all eigenvalues infinite breaks calculations. The correct conclusion from this is that we have a defective (wrong) Hamiltonian. This is not a big deal, nobody expects canonical quantization to always give correct results. We have a system with infinite number of degrees of freedom, not a finite set of harmonic oscillators, so maybe the Hamiltonian is something different than what we have for simple harmonic oscillator. But this does mean we should at least look for a better Hamiltonian (if not for a completely different formulation of the theory).
However, there are several different ways to improve on the above Hamiltonian, here are two of them:
high frequency cutoff in the above sum applied to zero point term, so the Hamiltonian has finite eigenvalues;
so-called normally ordered Hamiltonian, which has no zero-point term in any mode, so the Hamiltonian has finite eigenvalues, and zeroth eigenvalue is zero, so ground state has zero energy;
...
The first way introduces a modification of the Hamiltonian which "cuts off" (suppresses to zero) contribution of terms with frequency that is higher than some arbitrary cutoff frequency $\omega_c$; however, many calculations' results consistent with experiments are affected by changes in $\omega_c$ in a negligible way, as long as we don't put $\omega_c$ too low. When this cutoff frequency is selected in a particular way, connected to gravity constant (which is just additional assumption, foreign to QFT), then we get a definite but very high value for the zeroth eigenvalue, so called ground state energy or zero point energy. Then some people think, maybe there is a non-zero contribution due to ground state of quantum fields to 00 component of stress-energy in vacuum, and maybe we can calculate value of this contribution, and then use this result to interpret the non-zero value of the cosmological term in Einstein's equations or universe models, as being due to non-zero ground state energy of quantum fields. The cosmological term is actually not a part of the stress-energy tensor proper, but formally can be put into it to get the "effective stress tensor".
But, it is well-known that this assumption-loaded procedure (calculation and interpretation) produces energy density value that is 120 orders greater compared to the observed value of the effective energy density due to cosmological constant term. This bad result should give us pause and realize the procedure is obviously quite dubious right from the start, irrespective of the incorrect result: there is no good trusted theory involving gravity where such cutoff connected to gravity constant has sense, so there is no good reason to introduce such cutoff. Some people talk about this procedure as if it was a prediction of the value of cosmological constant, which it is not. Sabine Hossenfelder put it well: let me ask a rhetorical question: which theory was falsified and rejected as a result of observations showing this "predicted" value is completely wrong? None, because it is obvious the above procedure is full of dubious assumptions. What is actually falsified is the dubious procedure, but since it is not an inescapable implication of QFT, nothing about QFT itself is falsified.
There is the second way to fix the original problem - just use the Hamiltonian without the zero point energy terms:
$$ H_{no} = \sum_{\mathbf k,\lambda} \hbar\omega_{\mathbf k} a_{\mathbf k,\lambda}^+ a_{\mathbf k,\lambda}~~~. $$ Sometimes this is called Hamiltonian in normal ordering. The lowest eigenvalue of this is zero - the most plausible value ever for the lowest possible energy of EM field in a theory where the concept of energy is based on the Poynting expressions. Expectation values for any non-pathological quantum state are finite, no infinite expectation values appear. No arbitrary cutoff, no heuristics with gravity constant, just a Hamiltonian-based theory of EM field, which correctly reproduces many experiments involving matter-EM field interaction. BUT, since the field ground state energy is zero, the theory cannot be connected to the concept of non-zero cosmological term in vacuum in any direct way. Non-zero cosmological constant term is really a concept from GR and GR-based cosmological models; there is no real logical necessity or need to have this GR concept linked to field ground state energy in QFT. I should say many people don't agree on this point, but none of them know how to formulate a functional QFT of gravity where the above procedure would be consistent both with that theory and observation of cosmological constant.