Hi
Different people have different ways of working, but certainly a beat that doesn't suit your piece will change the feel. Beats are incorrectly emphasized and that fights with the track you record, making performances more hesitant and lacking the energy of conviction.
When I record I put together a beat that goes with the song as early in the process as possible... ie one of the first steps after recording a guide guitar and vocal or piano and vocal, sometimes recorded using a click track playing in my headphones. I then construct the beat against that guide track. Even if I was using loops I would still start with that simple guide. I know it's not going to make it to the final mix, it's just a guide for recording the other tracks. Sometimes I might just record a guide vocal and the click itself.
Why the metronome/click? For me that is more related to synchronising with midi instrumentation etc. All real instruments is another matter. Bands communicate visually as well as through the music. Well rehearsed bands can "go with the mood" using subtle variations in tempo that after the fact can be difficult to recreate without the rest of the instruments playing... that is why back in the pre digital music days bands often made guide takes with the whole band playing before using the benefits of multitracks for overdubs and individual instrument replacement. Recordings that are made using the whole band playing certainly give a more authentic sound, but generally there are problems with isolation of sounds when doing that, especially in budget studios. In other words the bass spilling onto the guitar track... if you want to replace the bass it becomes harder to do as some of the old bass is present on other tracks!
The fact is it is easier to deal with adding sequenced digital tracks if the real instruments have played to click. That simple fact has added to the lack of tempo variation in modern music as a means to communicate feeling through music, probably more than any other factor. In contemporary music there is very little accelerando/deccelerando. This isn't always compatible with some music genres as it robs too much "feel".
So, what to do... to a great extent depends on what you want to end up with and how you are going to provide it. For example, ideally you might want a real symphony orchestra to playing accompanyment, in reality you are more likely to play those parts with a synth. If that is a detailed arrangement that could be numerous instruments arranged in a sequencer.... and here you run into the need for the real instruments to play to a click track.
You can of course program accelerando or deccelerando in midi... it's just not as straight forward. You can also record midi tracks heedless of the precise tempo, varying with the real instruments... but just wait until you want to print a score, or edit the midi notes... and you will instantly think "I wish I had used a click track"!
So the answer is you have to be flexible in your approach but think ahead to the needs of the song and it's arrangement, and plan your recording accordingly.
In days of the past studio time was expensive, so bands generally had to practice and experiment in rehearsals... now with the profusion of home studios of a good quality bands/writers can afford more experimentation in the recording studio and then (depending on the needs of the recording) take the results of the home experiments into the studio and use that as a starting or reference point.
Certainly only the rich or well sponsored bands could afford to go into a studio unprepared.
A bit of a ramble but I hope I made sense!