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. 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 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 quite dubious right from the start, even before it produces obviously incorrect result: there is no good trusted theory involving gravity where such cutoff has sense, so there is no good reason to introduce cutoffs connected to gravity constant. Some people talk about this procedure with very wrong result as a prediction of the 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 everybody knows the above procedure is full of dubious assumptions. What is actually falsified is the dubious procedure, but since it is not actually a serious prediction 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 it is called the Hamiltonian in normal ordering. The lowest eigenvalue of this is zero - the most plausible value ever, for energy of a ground state. Expectation values for any state are finite, no infinite expectation values. No arbitrary cutoff, no heuristics with gravity constant, just a Hamiltonian based theory of EM field, which reproduces most results consistent with experiments. BUT, with this, there is no direct way to get non-zero vacuum energy, and then obviously the theory cannot be connected to the concept of cosmological constant in any direct way. Cosmological constant is really a concept from GR and GR-based cosmological models, and there need not be any connection to spurious zero point terms in QFT.