Let the campaign be a small part of a bigger picture.
It is possible to run a mostly "pulp" campaign into high tiers, but you may need to put in quite a bit of work into homebrew.
I have run a campaign like this into tier 4, where "pulp" was a regular feature. I would have happily run it right up to level 20 but we had to fold because it was an in-person campaign and several people moved away.
The higher-end published campaigns often set the story so that the PCs can potentially become the heroes of the story and sometimes even "save the world".
In our campaign there was a potential cataclysm but that was the story line of the world setting itself.
The PCs realised early on that the were a speck of sand in this vast setting as the the creatures who were at odds with each other where like demi gods and their struggle with each other was ancient spanning to times immemorial. In fact, they were the same creature with different aspects: one sub-race aligning themselves with life, the other with death. I built quite a bit of fun lore into this.
This meant that there were communities of beings in the world setting siding with one or the other. But, there were also others that were quite neutral and seeing that both were necessary and that it all as an on-going natural struggle. They met this view when they came across a "copse" of hermit druids.
To give an example, the Merfolk in one region and the Kuo-Toa were at odds. The PCs chose to help out the Merfolk village to take down a Kuo-Toa high priest in a small part of the world. They decided that was their main goal. It was quite organic as I gave them various options to follow and was creating content as we went along - in between the sessions mostly.
Along the way, I set various other stories where, for instance, they helped a Halfling that got lost at sea, liberated a little tribe of Flumph from a young blue dragon, investigated why the villages in a region were all "going werewolf". There was a lot of pulp - in fact, that was the main part of the campaign.
During their first 5 levels, they saw the bigger picture and realised that the story line would happen over millennia and had been happening for much longer than that. They would not see the end result of this as their PCs would long be dead, gone and forgotten by then. So, they relaxed into the campaign and made choices based on where their curiosity took them. It took quite a bit of work and not all my "pulp" mini stories were followed - which was ok. Sometimes, I just adapted them into for higher levels.
I had a couple of players that enjoyed working things out, so I built detective story lines, complex traps, puzzles and riddles. In the mini story line about the Flumphs, I introduced a magical barrier in a big cave system with an unknown language etched in monoliths. To pass they worked out they had to read out the coded message. So they went off to explore the monoliths and worked out the secret alphabet. They decoded the message. But, to pass the barrier they had to say it out loud. So I got them to actually read it out loud. I purposely turned it into a fun tongue twister. There was a time pressure too, as once they activated the magical code they only had a few minutes to get through or else the gas cloud would kill the PCs. I used a sand clock, turned it and said: you have 2 minutes to get through. It was a lot of fun.
I tend to go back to the Know Your Players section in the Introduction of the Dungeon Master's Guide. I asked them out-of-game what sort of adventures they liked. Some told me later on because they were new to D&D and I adjusted accordingly my prep for our next play sessions.
I think the way we had to orgainse our meet-ups helped in some way, too. We all had quite a few life commitments so we could only meet once per month - but for a whole weekend. We played 8-10 hours both on Saturday and Sunday and had lunch, snack and dinner breaks. I think this helped as I had a month to create content and stories.
There was also a fair amount of spontaneous deviation from the story line, so they kept me on my toes. It helped that I asked them to keep track of the story lines. At the beginning of the play sessions, the scribe read out a summary of what had happened in the last session and we went from there.
On a practical note, I've found that for homebrews it can be helpful to use a random adventure generator to generate ideas. I used the donjon website. I did not follow what was mentioned to the "T" but it worked well. I normally did have to tone it down though, e.g. if it talked about "an epic battle of clashing armies", I made it about helping out a small squad by getting them some food supplies.