That will undermine trust in a DM, absolutely. Nothing screams "I just changed this rule because it messes with what I want to happen in the game" like houseruling on the fly.
Yeah. If a rule or house rule proves problematic in play, I typically let it stand but then warn the players, "I think this is problematic result. I may revisit this rule and alter it at some point in the future." Changing a rule mid-session is just a bad idea unless everyone at the table agrees to a rule change because no one is happy with the rule.
Honestly, I love old school D&D (and I love 5e) and even I'm getting tired of that constant refrain.
I ran 3e with deliberate 1e feel. But I would never want to go back and run straight up 1e AD&D. And while I understand the people who are running cleaned up and modernized versions of AD&D and BECMI, I'd never switch to that either. As hard of work as it was to get 3e to run like I wanted 1e to run, it would be even more work to get 1e to run the way I'd want 1e to run.
Every single edition of the game has had problems because there is no such thing as a perfect ruleset. There are only tradeoffs. And on top of the tradeoffs, no rule set is ever perfect either. Every edition needs cleanup and extension because every designer had a deadline to meet and made mistakes. Every edition of the game did something well and something badly. You can talk about the tradeoffs, but ain't no sense in fighting over which is best.