Saving more is not a bad idea. If you have a contingency plan for job loss that you haven’t discussed that would be important to going ahead with buying now. If you don’t have some back-up and/or just can’t, or won’t, wait long enough to save more, buy a cheaper house or at least a house that is as near certain to cash flow as is possible, in the case you can’t pay for it .
Nobody ever mentions it here, but many times family is backing the risk and encouraging buying. Might or might not be the case with Emily.