I'll be the first to say that ordinal regression is a vaguely understood term. I've seen the term thrown around in the context of regression in which the outcome is ordinal, and/or when the regressor / predictor of interest is ordinal. Generally when one says they do __ regression, the modifier refers to a specific mean / variance relationship and transformation (or general formulation) of the outcome variable as dictated by a generalized linear model (GLM). An example of that is logistic regression, relative risk regression, proportional hazards regression, etc.
There are a whole host of models that are adequate for analyzing ordinal outcomes, none of which should be specifically referred to as "ordinal regression". Doing as you've done and treating numeric ordered levels as outcomes is a perfectly adequate way of analyzing such data. The beta coefficients are interpreted as an expected difference in the group-average Likert response comparing groups differing by 1 unit in the predictor of interest.
An alternative approach is a cumulative link model. In particular, the proportional odds model is often used for analyzing Likert or ordinal outcomes. This puts substantially more influence on top box / bottom box responses, but mostly gives results consistent with linear regression. The results are also fairly consistent with what you'd get from splitting outcomes into high / low and treating them as binary outcomes in logistic regression. That's exactly what PropOdds models do, in fact. They incrementally threshold responses and estimate model parameters borrowing information across each possible threshold response. The beta coefficients are interpreted as the log-odds of reporting an incrementally higher Likert response comparing groups differing by 1 unit in the predictor of interest. This model should be assessed for the proportional odds assumption to verify that there is not a tremendous violation of regularity conditions.
What's not appropriate for analyzing outcomes of ordinal nature is treating them as categorical. I saw this in the analysis of stage-at-presentation data in cancer patients randomized to receive either a novel screening service versus standard of care. They found significant results looking at the treated versus control frequencies of all possible cancer stages, but fitting either linear or proportional odds models did not find significance. When you looked at frequencies, what you saw was that there was a marginal difference in the number of people in intermediate cancer stages, balanced by marginal differences in very low or very high cancer stages. I argued very hard that this was evidence that, in the long run, the intervention had little evidence to "improve the overall stage at presentation" in the population. If the outcome is ordered, you absolutely must use a technique which borrows information effectively across response levels.