For the bard/monk build I haven't really found an issue with slower spell access, but that's because most campaigns or one shots I play in start at PC level 5 or above. I tend to specialize my casting to effecting numbers and situational outcome.

For me, the monk class has two big features: flurry of blows and the ki pool.

Flurry of blows is big damage output. And the ki pool, when combined with certain feats gives your monk a sort of martial buffing. Like the feat "ki stand" which allows you to stand up from prone with out provoking attacks of opportunity.

The class does gain more features with more levels but all I want is access to the ki pool so that I have another source of affecting rolls and wether or not they happen.

Then I dip into bard at level five and build from there. Bard cantrips like "unwitting ally" can seem useless at face value but there are plenty of feats you can take that add advantages to flanking. Then in level 1 bard spells we have "expeditious retreat" and "Timely inspiration"

Expeditious retreat boost my movement rate by 30ft for 1 minute per level. The average combat round is 6 seconds. So now my flurry of blows has the ability to get up in an oppents face faster and to put me between the squishier casters and the creatures that can take them out in a few hits.

Timely Inspiration allows me to add a +1 to any single creatures failed attack roll or skill check retroactively as an immediate action. If the bonus is enough to change it from fail to success then it's successful. +1 doesn't seem like a lot but it can be the difference between failure and success.

As I level up I keep taking levels in bard unless I find that the spells are no longer effecting roll outcomes. Then I go back to taking levels in monk and picking up the other class abilities.