Here is an approach that leverages lightning/uiRecordApi to get all relevant data and does not rely on a lightning-record-edit/view-form.

@op: please ignore fieldNames in screenshot :)

XML

<?xml version="1.0" encoding="UTF-8" ?>
<LightningComponentBundle xmlns="http://soap.sforce.com/2006/04/metadata">
    <apiVersion>48.0</apiVersion>
    <isExposed>true</isExposed>
    <masterLabel>Section with Fields</masterLabel>
    <targets>
        <target>
            lightning__RecordPage
        </target>
    </targets>
    <targetConfigs>
        <targetConfig targets="lightning__RecordPage">
            <property name="fieldNames" label="Field Names" type="String" required="true" default="Name,Industry,NumberOfEmployees" description="Commma seperated list of fully qualified API fieldnames that should be displayed in this section" />
        </targetConfig>
    </targetConfigs>
</LightningComponentBundle>

HTML

<template>
  <lightning-card title="Get Field Values" icon-name="standard:question_best">
    <div class="slds-card__body_inner">
      <h2 class="slds-text-heading_small">
        Current Settings
      </h2>
      <p>
        recordId: {recordId} / objectApiName: {objectApiName} /
        configuredFields: {fieldNames}
      </p>
      <h2 class="slds-text-heading_small slds-var-m-top_medium">
        Field Values
      </h2>
      <template for:each={results} for:item="field">
        <p key={field.fieldName}>{field.fieldName}: {field.value}</p>
      </template>
    </div>
    <template if:true={errorMessage}>
      <pre>{errorMessage}</pre>
    </template>
  </lightning-card>
</template>

JS

import { LightningElement, api, wire } from "lwc";
import { getRecordUi } from "lightning/uiRecordApi";

export default class RecordpageSection extends LightningElement {
  @api
  fieldNames;

  @api
  recordId;

  @api
  objectApiName;

  apiFieldnames = [];

  errorMessage;

  results = [];

  connectedCallback() {
    this.fieldNames.split(",").forEach(fieldName => {
      this.apiFieldnames.push(this.objectApiName + "." + fieldName);
    });
  }

  @wire(getRecordUi, {
    recordIds: "$recordId",
    layoutTypes: "Compact",
    modes: "View",
    optionalFields: "$apiFieldnames"
  })
  recordInformation({ error, data }) {
    if (data) {
      let fieldInfo = data.records[this.recordId].fields;
      let matchingFields = [];
      Object.keys(fieldInfo).forEach(fieldName => {
        let qualifiedFieldname = this.objectApiName + "." + fieldName;
        //originalIndex is used to sort results as configured
        let originalIndex = this.apiFieldnames.indexOf(qualifiedFieldname);
        if (originalIndex > -1) {
          matchingFields[originalIndex] = {
            fieldName: fieldName,
            value: fieldInfo[fieldName].value
          };
        }
      });
      this.results = matchingFields;
    } else if (error) {
      this.errorMessage = JSON.stringify(error);
    }
  }
}