Your linear regression can't predict on the missing data if it doesn't have a predictor. So your value is not imputed.
Although it does involve regressions, Multivariate Imputation by Chained Equations (MICE) is a bit different from your linear regression approach. In a nutshell, missing variables are first tentatively filled, which makes them suitable as predictors, and then they are iteratively imputed. I would suggest looking at the pseudocode in Azur, M. J.; Stuart, E. A.; Frangakis, C. & Leaf, P. J. (2011) Multiple Imputation by Chained Equations: What is it and how does it work?. International journal of methods in psychiatric research, 20, 40-49 to understand what the algorithm does.
Short answer: your gut-feeling is right.
Longer answer: The strength of imputation lies in the pooling procedure. If you read the MICE manual, the writers go in depth about this. They state that imputation is not a technique that you apply to a dataset with missing data to complete the empty cells, but that it is a combination of setting up a strategy to replace missing data (using chained equations in the mice case), performing the analysis and subsequently pooling the result which answers your research question (i.e. the reason you performed the analysis). As such, these steps are mandatory.
Now, more specifically to your situation. In the original data with missings there might be data which is selectively missing. This could lead to bias. Moreover, most analyses require complete data on all variables, so you'll need to exclude cases or handle them in some way. Using imputation you complete the missing data with 'guestimates' based on the assumption your data is 'missing at random, conditional on known and observed data' (MAR). However, because these are guesses based on your data, you add some randomness and repeat this completion process multiple times in order to create a distribution of guesses.
If you would analyse these data in the 'long' format you mention, you'd have basically pumped up your sample's size with a factor equal to the number of imputation sets! This will undoubtedly increase the precision of your estimates. But, this is wrong! Cases which were complete from the start will have been copied and more importantly, you did not take into account the uncertainty of your guestimates.
The better way therefore is to analyze the data per imputation set. This way you get an 'm' amount of results. However, you do not know which imputation set is the 'most correct' (if there even is such a dataset). As such the average coefficient of all models is your best estimate for the 'true' estimate. For the precision (and hypothesis testing/confidence intervals) you then need to appropriately handle the standard error. Now the uncertainty comes into play. Using Rubins rules you average the standard errors of all and add 'a little extra' to represent the variation of the estimates across imputation sets.
Conclusion
Finally, creating your confidence intervals and performing your hypothesis tests using these pooling rules usually decreases bias and biased inferences compared to a complete case analysis. Compared to your long-format dataset, the coefficient might be pretty similar, but as you and your gut-feeling rightly pointed out, the results are way to precise (too narrow confidence intervals; too low p-values) than can actually should be concluded from these imputation analyses.
Best Answer
If you have access to the data set the model was trained on, you could impute new data and then compare means, standard deviations etc. to see how they differ.
You could also work backwards and use the model as is on a data set then compute statistics for that set, then try out different imputation techniques on the test data set and continue to generate statistics and compare and contrast them.
If predictor variables are missing, you might be able to throw out those data points if you have a sufficiently large enough data set to work of off. Also if this is the case, you could retrain the model using imputation and cross validation to achieve a desirable prediction score.
Remember if you do retrain the model on a new data set and there is missing data that you wish to impute, perform cross validation and split your data set into test/train sets before imputation. As this will mimic real life. Then perform imputation on the training set and once you have the trained model you want with the type of imputation you want. Perform that same data preprocessing on your testing set.
Hope this helps point you in the right direction!